Test bank for Precalculus 6th Edition by Robert F. Blitzer

TEST BANK FOR PRECALCULUS 6TH EDITION BY ROBERTF. BLITZER Ch. 2 Polynomial and Rational Functions 2.1 Complex Numbers 1 Add and Subtract Complex Numbers MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Add or subtract as indicated and write the result in standard form. 1) (5 – 6i) + (9 + 9i) A) 14 + 3i B) 14 – 3i C) -4 + 15i D) -14 – 3i 2) (9 + 5i) – (-8 + i) A) 17 + 4i B) 17 – 4i C) 1 + 6i D) -17 – 4i 3) 8i + (-7 – i) A) -7 + 7i B) -7 + 9i C) 7 – 7i D) 7 – 9i 4) 5i – (-5 – i) A) 5 + 6i B) -5 – 6i C) 5 – 4i D) -5 + 4i 5) (-7 + 6i) – 9 A) -16 + 6i B) 16 – 6i C) 2 + 6i D) 2 – 6i 6) -2 – (- 2 – 8i) – (- 2 – 6i) A) 2 + 14i B) 2 – 14i C) 4 – 14i D) 4 + 14i 7) (4 – 3i) + (1 – 6i) + (4 + 5i) A) 9 – 4i B) 7 + 8i C) 1 – 14i D) 5 – 9i 2 Multiply Complex Numbers MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the product and write the result in standard form. 1) -5i(3i – 9) A) 15 + 45i B) -15 + 45i C) 45i – 15i2 D) 45i + 15i2 2) 7i(-3i + 6) A) 21 + 42i B) -21 + 42i C) 42i – 21i2 D) 42i + 21i2 3) (9 + 8i)(5 – 5i) A) 85 – 5i B) 85 + 5i C) 5 + 85i D) -40i2 – 5i + 45 4) (-6 + 9i)(5 + i) A) -39 + 39i B) -21 + 39i C) -39 – 51i D) -21 – 51i 5) (8 – 6i)(-5 – 3i) A) -58 + 6i B) -58 – 54i C) -22 + 6i D) -22 – 54i 6) (9 + 3i)(9 – 3i) A) 90 B) 81 – 9i2 C) 72 D) 81 – 9i Page 1 7) (-4 + i)(-4 – i) A) 17 B) -4 C) 16 D) -15 8) (4 + 9i)2 A) -65 + 72i B) 97 + 72i C) -65 D) 16 + 72i + 81i2 Perform the indicated operations and write the result in standard form. 9) (7 + 8i)(3 – i) – (1 – i)(1 + i) A) 27 + 17i B) 31 + 17i C) 29 + 17i D) 27 + 31i 10) (2 + i)2 – (6 – i)2 A) -32 + 16i B) 32 + 16i C) -48 D) -32 – 16i Complex numbers are used in electronics to describe the current in an electric circuit. Ohm’s law relates the current in a circuit, I, in amperes, the voltage of the circuit, E, in volts, and the resistance of the circuit, R, in ohms, by the formula E = IR. Solve the problem using this formula. 11) Find E, the voltage of a circuit, if I = (8 + 9i) amperes and R = (4 + 7i) ohms. A) (-31 + 92i) volts B) (-31 – 92i) volts C) (92 – 31i) volts D) (92 + 31i) volts 12) Find E, the voltage of a circuit, if I = (18 + i) amperes and R = (2 + 3i) ohms. A) (33 + 56i) volts B) (33 – 56i) volts C) (-18 + 56i) volts D) (-18 – 56i) volts 3 Divide Complex Numbers MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Divide and express the result in standard form. 1) 7 8 – i A) 56 65 + 7 65 i B) 56 65 – 7 65 i C) 89 + 19i D) 89 – 19 i 2) 8 2 + i A) 16 5 – 85 i B) 16 5 + 85 i C) 16 3 + 83 i D) 16 3 – 83 i 3) 10i 3 + i A) 1 + 3i B) -1 + 3i C) 1 + 10i D) 1 – 3i 4) 4i 3 + i A) 2 5 + 65 i B) – 25 + 65 i C) 1 2 + 32 i D) 2 5 – 65 i 5) 2i 1 + 7i A) 7 25 + 1 25 i B) 1 25 + 7 25 i C) – 7 24 + 1 24 i D) – 1 24 – 7 24 i Page 2 6) 4 + 5i 5 – 4i A) i B) -i C) 1 D) -1 7) 5 – 4i 8 + 6i A) 4 25 – 31 50 i B) 27 – 31 28 i C) 32 25 + 1 25 i D) 16 7 – 31 28 i 8) 3 + 4i 9 – 3i A) 16 + 12 i B) 1 72 + 1 24 i C) 13 2 – 92 i D) 13 24 + 1 24 i 9) 2 + 3i 5 + 2i A) 16 29 + 11 29 i B) 16 21 + 11 21 i C) 4 29 – 19 29 i D) 4 21 + 11 21 i 10) 5 + 8i 4 + 2i A) 95 + 11 10 i B) 3 2 + 11 12 i C) 2 5 – 21 5 i D) 13 + 11 12 i 11) 4 – 3i 5 – 3i A) 29 34 – 3 34 i B) 29 16 – 3 16 i C) 11 34 + 27 34 i D) 11 16 – 3 16 i 4 Perform Operations with Square Roots of Negative Numbers MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Perform the indicated operations and write the result in standard form. 1) -16 + -81 A) 13i B) -13i C) 36i D) -13 2) -3 – -121 A) i( 3 – 11) B) 3i – 11 C) 3i – 11i D) i( 3 + 11) 3) 3 -64 + 4 -49 A) 52i B) -52 C) 52 D) -52i 4) 2 -32 + 5 -50 A) 33i 2 B) -33 2 C) 33 2 D) -33i 2 5) (-8 – -49)2 A) 15 + 112i B) 113 + 112i C) 64 + 49i D) 64 – 49i Page 3 6) (-6 + -100)2 A) -64 – 120i B) 136 + 120i C) 36 + 100i D) 36 – 100i 7) ( 6 – – 64)( 6 + – 64) A) 70 B) -58 C) 6 – 64i D) 6 – 8i 8) (3 + -3) (3 + -2) A) (9 – 6 )+ (3 2 + 3 3)i B) (9 + 6 )- 15i C) 3 – 6 6i D) 15 + 36i 9) -2 + -12 2 A) -1 + i 3 B) -1 – i 3 C) 1 + i 3 D) -1 + i 2 10) -42 – -252 6 A) -7 – i 7 B) -7 + i 7 C) 7 + i 7 D) -7 – i 6 11) -16(5 – -9) A) 12 + 20i B) 20i – 12 C) 20i – 12i2 D) 20i + 12i2 12) ( -9)( -64) A) -24 B) 24i2 C) 24 D) -24i 5 Solve Quadratic Equations with Complex Imaginary Solutions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the quadratic equation using the quadratic formula. Express the solution in standard form. 1) x2 + x + 2 = 0 A) – 12 ± i 7 2 B) 12 ± i 7 2 C) 12 ± 7 2 D) – 12 ± 7 2 2) x2 – 12x + 40 = 0 A) {6 ± 2i} B) {6 ± 4i} C) {6 + 2i} D) {4, 8} 3) 8×2 + 3x + 3 = 0 A) – 3 16 ± i 87 16 B) – 3 16 ± 87 16 C) 3 16 ± i 87 16 D) 3 16 ± 87 16 4) 16×2 – 5x + 1 = 0 A) 5 32 ± i 39 32 B) – 5 32 ± i 39 32 C) – 5 32 ± 39 32 D) 5 32 ± 39 32 5) 5×2 = -9x – 7 A) – 9 10 ± i 59 10 B) – 9 10 ± 59 10 C) 9 10 ± i 59 10 D) 9 10 ± 59 10 Page 4 2.2 Quadratic Functions 1 Recognize Characteristics of Parabolas MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a quadratic function is given. Determine the function’s equation. 1) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) f(x) = (x + 3)2 + 3 B) g(x) = (x + 3)2 – 3 C) h(x) = (x – 3)2 + 3 D) j(x) = (x – 3)2 – 3 2) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) g(x) = (x + 3)2 – 3 B) f(x) = (x + 3)2 + 3 C) h(x) = (x – 3)2 + 3 D) j(x) = (x – 3)2 – 3 3) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) h(x) = (x – 2)2 + 2 B) g(x) = (x + 2)2 – 2 C) f(x) = (x + 2)2 + 2 D) j(x) = (x – 2)2 – 2 Page 5 4) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) j(x) = (x – 2)2 – 2 B) g(x) = (x + 2)2 – 2 C) h(x) = (x – 2)2 + 2 D) f(x) = (x + 2)2 + 2 5) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) f(x) = x2 – 4x + 4 B) g(x) = x2 + 4x + 4 C) h(x) = x2 – 2 D) j(x) = x2 + 2 6) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) g(x) = x2 + 2x + 1 B) f(x) = x2 – 2x + 1 C) h(x) = x2 – 1 D) j(x) = x2 + 1 Page 6 7) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) h(x) = x2 – 1 B) g(x) = x2 + 2x + 1 C) f(x) = x2 – 2x + 1 D) j(x) = x2 + 1 8) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) j(x) = x2 + 3 B) g(x) = x2 + 6x + 9 C) h(x) = x2 – 3 D) f(x) = x2 – 6x + 9 9) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) j(x) = -x2 + 2 B) g(x) = -x2 + 4x + 4 C) h(x) = -x2 – 2 D) f(x) = -x2 – 4x – 4 Page 7 10) -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) h(x) = -x2 – 1 B) g(x) = -x2 + 2x + 1 C) j(x) = -x2 + 1 D) f(x) = -x2 – 2x – 1 Find the coordinates of the vertex for the parabola defined by the given quadratic function. 11) f(x) = (x + 1)2 + 1 A) (-1, 1) B) (-1, -1) C) (0, 1) D) (1, 0) 12) f(x) = x2 + 8 A) (0, 8) B) (-8, 0) C) (0, -8) D) (8, 0) 13) f(x) = (x + 5)2 + 9 A) (-5, 9) B) (-9, 5) C) (9, -25) D) (9, -5) 14) f(x) = 9 – (x + 5)2 A) (-5, 9) B) (5, 9) C) (9, 5) D) (9, -5) 15) f(x) = (x + 3)2 – 2 A) (-3, -2) B) (3, 2) C) (3, -2) D) (-3, 2) 16) y + 4 = (x + 2)2 A) (- 2, – 4) B) (2, – 4) C) (4, – 2) D) (4, 2) 17) f(x) = 11(x – 5)2 + 9 A) (5, 9) B) (11, 5) C) (-5, 9) D) (9, -5) 18) f(x) = -7(x – 2)2 – 8 A) (2, -8) B) (-8, 2) C) (-2, -8) D) (-7, -2) 19) f(x) = x2 – 8 A) (0, -8) B) (1, 0) C) (0, 8) D) (8, 0) 20) f(x) = x2 + 12x – 1 A) (-6, -37) B) (6, 107) C) (12, 287) D) (-6, -109) 21) f(x) = -x2 + 14x + 8 A) (7, 57) B) (-7, -139) C) (14, 8) D) (-7, -41) Page 8 22) f(x) = 8 – x2 + 2x A) (1, 9) B) (- 1, 9) C) (1, – 9) D) (- 1, – 9) 23) f(x) = -6×2 – 12x + 4 A) (-1, 10) B) (1, -14) C) (-2, -8) D) (2, -44) Find the axis of symmetry of the parabola defined by the given quadratic function. 24) f(x) = x2 + 7 A) x = 0 B) x = 7 C) x = -7 D) y = 7 25) f(x) = (x + 3)2 + 7 A) x = -3 B) x = 3 C) y = 7 D) y = -7 26) f(x) = 6 – (x + 3)2 A) x = -3 B) x = 3 C) x = 6 D) x = -6 27) f(x) = (x + 1)2 – 9 A) x = -1 B) x = 1 C) x = -9 D) x = 9 28) y + 9 = (x – 3)2 A) x = 3 B) x = -3 C) y =9 D) y = -9 29) f(x) = 11(x – 3)2 + 7 A) x = 3 B) x = 11 C) x = -3 D) x = 7 30) f(x) = -7(x – 4)2 – 6 A) x = 4 B) x = -6 C) x = -4 D) x = -7 31) f(x) = x2 + 12x – 5 A) x = -6 B) x = 6 C) x = 12 D) x = -41 32) f(x) = -x2 + 2x – 3 A) x = 1 B) x = -1 C) x = 2 D) x = -2 33) f(x) = 2×2 + 4x – 7 A) x = -1 B) x = 1 C) x = -2 D) x = -9 Find the range of the quadratic function. 34) f(x) = x2 + 6 A) [6, ∞) B) (-∞, 6] C) [-6, ∞) D) [0, ∞) 35) f(x) = (x + 4)2 + 8 A) [8, ∞) B) [-8, ∞) C) [4, ∞) D) [-4, ∞) 36) f(x) = 4 – (x + 2)2 A) (-∞, 4] B) [4, ∞) C) (-∞, 2] D) [-2, ∞) 37) f(x) = (x + 8)2 – 3 A) [-3, ∞) B) (-∞, -8] C) (-∞, -3] D) [-8, ∞) Page 9 38) y + 4 = (x – 2)2 A) [- 4, ∞) B) (-∞, – 2] C) [4, ∞) D) (-∞, 4] 39) f(x) = 11(x – 4)2 + 5 A) [5, ∞) B) [4, ∞) C) (-∞, 5] D) [-5, ∞) 40) f(x) = -7(x – 5)2 – 7 A) (-∞, -7] B) (-∞, 5] C) [-7, ∞) D) [-5, ∞) 41) f(x) = x2 – 8x – 8 A) [-24, ∞) B) [-4, ∞) C) (-∞, -24] D) (-∞, -56] 42) f(x) = -x2 + 10x + 3 A) (-∞, 28] B) [28, ∞) C) [5, ∞) D) (-∞, 5] 43) f(x) = 2×2 + 3x – 9 A) [- 81 8 , ∞) B) (-∞, – 81 8 ] C) [- 34 , ∞) D) (-∞, – 34 ] 44) f(x) = -3×2 – 6x A) (-∞, 3] B) (-∞, – 3] C) (-∞, – 1] D) (-∞, 1] Find the x-intercepts (if any) for the graph of the quadratic function. 45) f(x) = x2 – 4 A) (-2, 0) and (2, 0) B) (-4, 0) C) (2, 0) D) No x-intercepts 46) f(x) = (x – 1)2 – 1 A) (0, 0) and (2, 0) B) (0, 0) and (-2, 0) C) (0, 0) and (-1, 0) D) (-2, 0) and (2, 0) 47) y + 4 = (x – 2)2 A) (0, 0) and (4, 0) B) (0, 0) and (-4, 0) C) (-4, 0) and (4, 0) D) (0, 0) 48) f(x) = 4 + 5x + x2 A) (-1, 0) and (-4, 0) B) (1, 0) and (4, 0) C) (1, 0) and (-4, 0) D) (-1, 0) and (4, 0) 49) f(x) = x2 + 18x + 67 Give your answers in exact form. A) (-9 ± 14, 0) B) (9 + 14, 0) C) (9 ± 67, 0) D) (-18 ± 67, 0) 50) f(x) = -x2 + 11x – 30 A) (5, 0) and (6, 0) B) (-5, 0) and (-6, 0) C) (5, 0) and (-6, 0) D) No x-intercepts 51) f(x) = 2×2 – 9x + 10 A) (2, 0) and (2.5, 0) B) (2, 0) and (-2.5, 0) C) (5, 0) and (1, 0) D) (5, 0) and (- 1, 0) 52) f(x) = 2×2 + 26x + 72 A) (-4, 0) and (-9, 0) B) (4, 0) and (9, 0) C) (-4, 0) and (9, 0) D) (4, 0) and (-9, 0) Page 10 53) 3×2 + 6x + 2 = 0 Give your answers in exact form. A) -3 ± 3 3 , 0 B) -3 ± 3 6 , 0 C) -6 ± 3 3 , 0 D) -3 ± 15 3 , 0 Find the y-intercept for the graph of the quadratic function. 54) f(x) = -x2 – 2x + 8 A) (0, 8) B) (8, 0) C) (0, -4) D) (0, -8) 55) y + 9 = (x – 3)2 A) (0, 0) B) (0, -6) C) (0, 6) D) (9, 0) 56) f(x) = 4 + 5x + x2 A) (0, 4) B) (0, 1) C) (0, -4) D) (0, 5) 57) f(x) = x2 + 5x – 6 A) (0, -6) B) (0, 3) C) (0, 6) D) (0, 5) 58) f(x) = (x + 3)2 – 9 A) (0, 0) B) (0, 6) C) (0, 9) D) (0, -9) 59) f(x) = 4×2 – 3x – 7 A) (0, -7) B) (0, 7) C) 0, 74 D) 0, – 74 Find the domain and range of the quadratic function whose graph is described. 60) The vertex is (1, -14) and the graph opens up. A) Domain: (-∞, ∞) Range: [-14, ∞) B) Domain: [1, ∞) Range: [-14, ∞) C) Domain: (-∞, ∞) Range: (-∞, -14] D) Domain: (-∞, ∞) Range: [1, ∞) 61) The vertex is (-1, -10) and the graph opens down. A) Domain: (-∞, ∞) Range: (-∞, -10] B) Domain: (-∞, -1] Range: (-∞, -10] C) Domain: (-∞, ∞) Range: [-10, ∞) D) Domain: (-∞, ∞) Range: (-∞, -1] 62) The minimum is 0 at x = -1. A) Domain: (-∞, ∞) Range: [0, ∞) B) Domain: [-1, ∞) Range: [0, ∞) C) Domain: (-∞, ∞) Range: (-∞, 0] D) Domain: (-∞, ∞) Range: [-1, ∞) 63) The maximum is -8 at x = 1 A) Domain: (-∞, ∞) Range: (-∞, -8] B) Domain: (-∞, 1] Range: (-∞, -8] C) Domain: (-∞, ∞) Range: [-8, ∞) D) Domain: (-∞, ∞) Range: (-∞, 1] Solve the problem. 64) Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 11×2, but which has its vertex at (3, 4). A) f(x) = 11(x – 3)2 + 4 B) f(x) = 11(x + 3)2 + 4 C) f(x) = (11x + 3)2 + 4 D) f(x) = 11(x + 4)2 + 3 Page 11 65) Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 5×2, but which has a minimum of 7 at x = 3. A) f(x) = 5(x – 3)2 + 7 B) f(x) = 5(x + 3)2 + 7 C) f(x) = -5(x – 3)2 + 7 D) f(x) = 5(x + 7)2 – 3 66) Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = -7×2, but which has a maximum of 7 at x = 5. A) f(x) = -7(x – 5)2 + 7 B) f(x) = -7(x + 5)2 + 7 C) f(x) = 7(x – 5)2 + 7 D) f(x) = -7(x – 5)2 – 7 Page 12 2 Graph Parabolas MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the vertex and intercepts to sketch the graph of the quadratic function. 1) y – 4 = (x + 5)2 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 13 2) f(x) = -3(x – 6)2 – 3 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 14 3) f(x) = (x – 5)2 – 6 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 15 4) f(x) = 1 – (x + 1)2 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 16 5) f(x) = x2 + 6x + 5 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 17 6) f(x) = -x2 – 4x + 5 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 18 7) f(x) = x2 – 8x + 7 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 19 8) f(x) = – 6x + 8 + x2 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 20 9) f(x) = -x2 + 2x + 8 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 21 10) f(x) = 3 – x2 – 2x -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 22 11) f(x) = 4 + 5x + x2 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 23 12) f(x) = 3×2 + 24x + 47 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 10 x y 10 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 3 Determine a Quadratic Function’s Minimum or Maximum Value MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 1) f(x) = x2 – 2x – 2 A) minimum; 1, – 3 B) maximum; 1, – 3 C) minimum; – 3, 1 D) maximum; – 3, 1 2) f(x) = -x2 + 2x – 1 A) maximum; 1, 0 B) minimum; 1, 0 C) minimum; 0, 1 D) maximum; 0, 1 Page 24 3) f(x) = 4×2 – 2x – 2 A) minimum; 14 , – 94 B) maximum; 14 , – 94 C) minimum; – 94 , 14 D) maximum; – 94 , 14 4) f(x) = 4×2 + 8x A) minimum; – 1, – 4 B) maximum; – 1, – 4 C) minimum; 1, – 4 D) maximum; 1, – 4 5) f(x) = -5×2 – 15x A) maximum; – 32 , 45 4 B) minimum; – 32 , 45 4 C) minimum; 32 , – 45 4 D) maximum; 32 , – 45 4 4 Solve Problems Involving a Quadratic Function’s Minimum or Maximum Value MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) You have 300 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. A) 75 ft by 75 ft B) 150 ft by 150 ft C) 150 ft by 37.5 ft D) 77 ft by 73 ft 2) A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 296 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed? A) 10,952 ft2 B) 21,904 ft2 C) 5476 ft2 D) 16,428 ft2 3) You have 120 feet of fencing to enclose a rectangular region. What is the maximum area? A) 900 square feet B) 3600 square feet C) 14,400 square feet D) 896 square feet 4) You have 120 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. A) length: 60 feet, width: 30 feet B) length: 90 feet, width: 30 feet C) length: 60 feet, width: 60 feet D) length: 30 feet, width: 30 feet 5) A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross- sectional area and allow the greatest amount of water to flow. A) 4.5 inches B) 4 inches C) 5 inches D) 5.5 inches 6) A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 648 feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. A) 108 ft by 162 ft B) 162 ft by 162 ft C) 54 ft by 243 ft D) 81 ft by 162 ft 7) A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 600 feet of fencing is used. Find the maximum area of the playground. A) 15,000 ft2 B) 22,500 ft2 C) 11,250 ft2 D) 16,875 ft2 Page 25 8) The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function C(x) = 4×2 – 24x + 81. Find the number of automobiles that must be produced to minimize the cost. A) 3 thousand automobiles B) 6 thousand automobiles C) 45 thousand automobiles D) 12 thousand automobiles 9) In one U.S. city, the quadratic function f(x) = 0.0041×2 – 0.48x + 36.07 models the median, or average, age, y, at which men were first married x years after 1900. In which year was this average age at a minimum? (Round to the nearest year.) What was the average age at first marriage for that year? (Round to the nearest tenth.) A) 1959, 22 years old B) 1959, 50.1 years old C) 1936, 50.1 years old D) 1953, 36 years old 10) The profit that the vendor makes per day by selling x pretzels is given by the function P(x) = -0.002×2 + 1.6x – 200. Find the number of pretzels that must be sold to maximize profit. A) 400 pretzels B) 800 pretzels C) 0.8 pretzels D) 120 pretzels 11) The manufacturer of a CD player has found that the revenue R (in dollars) is R(p) = -5p2 + 1800p, when the unit price is p dollars. If the manufacturer sets the price p to maximize revenue, what is the maximum revenue to the nearest whole dollar? A) $162,000 B) $324,000 C) $648,000 D) $1,296,000 12) The owner of a video store has determined that the profits P of the store are approximately given by P(x) = -x2 + 20x + 51, where x is the number of videos rented daily. Find the maximum profit to the nearest dollar. A) $151 B) $100 C) $251 D) $200 13) The owner of a video store has determined that the cost C, in dollars, of operating the store is approximately given by C(x) = 2×2 – 28x + 730, where x is the number of videos rented daily. Find the lowest cost to the nearest dollar. A) $632 B) $338 C) $534 D) $828 14) The daily profit in dollars of a specialty cake shop is described by the function P(x) = -5×2 + 210x – 1600, where x is the number of cakes prepared in one day. The maximum profit for the company occurs at the vertex of the parabola. How many cakes should be prepared per day in order to maximize profit? A) 21 cakes B) 2205 cakes C) 441 cakes D) 42 cakes 15) Among all pairs of numbers whose sum is 56, find a pair whose product is as large as possible. A) 28 and 28 B) 14 and 14 C) 30 and 26 D) 55 and 1 16) Among all pairs of numbers whose difference is 26, find a pair whose product is as small as possible. A) -13 and 13 B) 13 and 13 C) -39 and -13 D) 39 and 13 17) An arrow is fired into the air with an initial velocity of 160 feet per second. The height in feet of the arrow t seconds after it was shot into the air is given by the function h(x) = -16t2 + 160t. Find the maximum height of the arrow. A) 400 ft B) 80 ft C) 1200 ft D) 720 ft Page 26 18) A person standing close to the edge on top of a 144-foot building throws a baseball vertically upward. The quadratic function s(t) = -16t2 + 64t + 144 models the ball’s height above the ground, s(t), in feet, t seconds after it was thrown. After how many seconds does the ball reach its maximum height? Round to the nearest tenth of a second if necessary. A) 2 seconds B) 5.6 seconds C) 208 seconds D) 1.5 seconds 19) April shoots an arrow upward into the air at a speed of 64 feet per second from a platform that is 30 feet high. The height of the arrow is given by the function h(t) = -16t2 + 64t + 30, where t is the time is seconds. What is the maximum height of the arrow? A) 94 ft B) 26 ft C) 64 ft D) 30 ft 20) An object is propelled vertically upward from the top of a 80-foot building. The quadratic function s(t) = -16t2 + 112t + 80 models the ball’s height above the ground, s(t), in feet, t seconds after it was thrown. How many seconds does it take until the object finally hits the ground? Round to the nearest tenth of a second if necessary. A) 7.7 seconds B) 0.7 seconds C) 3.5 seconds D) 2 seconds 2.3 Polynomial Functions and Their Graphs 1 Identify Polynomial Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the function is a polynomial function. 1) f(x) = 4x + 6×4 A) Yes B) No 2) f(x) = 5 – x3 6 A) Yes B) No 3) f(x) = 9 – 2 x3 A) No B) Yes 4) f(x) = x4 – 8 x5 A) No B) Yes 5) f(x) = 2 x3 – x2 – 7 A) No B) Yes 6) f(x) = -15×5 + 9x + 5 x A) No B) Yes 7) f(x) = πx5 + 6×4 + 3 A) Yes B) No Page 27 8) f(x) = x3/2 – x5 + 3 A) No B) Yes 9) f(x) = 5×7 – x5 + 43 x A) Yes B) No 10) f(x) = 4×3 + 5×2 – 4x-4 + 80 A) No B) Yes Find the degree of the polynomial function. 11) f(x) = -4x + 6×3 A) 3 B) 1 C) -4 D) 6 12) f(x) = 8 – x3 5 A) 3 B) – 15 C) 0 D) 8 13) f(x) = πx4 + 6×3 – 8 A) 4 B) 3 C) π D) 1 14) f(x) = 5x – x2 + 54 A) 2 B) 1 C) 5 D) -1 15) g(x) = -17×4 – 9 A) 4 B) 5 C) 0 D) -17 16) h(x) = -7x + 3 A) 1 B) 2 C) 0 D) -7 17) 14×3 + 5×2 – 2x + 3y4 + 2 A) 4 B) 3 C) 10 D) 14 18) f(x) = 11×3 – 7×2 + 4 A) 3 B) 6 C) -7 D) 11 Page 28 2 Recognize Characteristics of Graphs of Polynomial Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph shown is the graph of a polynomial function. 1) x y A) not a polynomial function B) polynomial function 2) x y A) polynomial function B) not a polynomial function 3) x y A) polynomial function B) not a polynomial function Page 29 4) x y A) polynomial function B) not a polynomial function 5) x y A) not a polynomial function B) polynomial function 6) x y A) not a polynomial function B) polynomial function Page 30 Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. 7) f(x) = 3×2 – x3 A) 0, touches the x-axis and turns around; 3, crosses the x-axis B) 0, crosses the x-axis; 3, crosses the x-axis; – 3, crosses the x-axis C) 0, touches the x-axis and turns around; 3, crosses the x-axis; – 3, crosses the x-axis D) 0, touches the x-axis and turns around; 3, touches the x-axis and turns around 8) f(x) = x4 – 9×2 A) 0, touches the x-axis and turns around; 3, crosses the x-axis; -3, crosses the x-axis B) 0, crosses the x-axis; 3, crosses the x-axis; -3, crosses the x-axis C) 0, touches the x-axis and turns around; 9, touches the x-axis and turns around D) 0, touches the x-axis and turns around; 9, crosses the x-axis 9) x5 – 21×3 + 80x = 0 A) 0, crosses the x-axis; 4, crosses the x-axis; -4, crosses the x-axis; 5, crosses the x-axis; – 5, crosses the x-axis B) 0, touches the x-axis and turns around; 4, crosses the x-axis; -4, crosses the x-axis; 5, crosses the x-axis; – 5, crosses the x-axis C) 0, crosses the x-axis; 16, touches the x-axis and turns around; 5, touches the x-axis and turns around D) 0, touches the x-axis and turns around; 16, touches the x-axis and turns around; 5, touches the x-axis and turns around 10) x4 + 7×3 – 44×2 = 0 A) 0, touches the x-axis and turns around; -11, crosses the x-axis; 4, crosses the x-axis B) 0, touches the x-axis and turns around; 11, touches the x-axis and turns around; -4, touches the x-axis and turns around C) 0, crosses the x-axis; -11, crosses the x-axis; 4, crosses the x-axis D) 0, touches the x-axis and turns around; 11, crosses the x-axis; -4, crosses the x-axis 11) f(x) = x3 + 10×2 + 33x + 36 A) -3, touches the x-axis and turns around; -4, crosses the x-axis. B) -3, crosses the x-axis; -4, touches the x-axis and turns around C) 3, crosses the x-axis; -3, crosses the x-axis; -4, crosses the x-axis. D) 3, crosses the x-axis; -3, touches the x-axis and turns around; -4, crosses the x-axis. Page 31 12) f(x) = (x + 1)(x – 3)(x – 1)2 A) -1, crosses the x-axis; 3, crosses the x-axis; 1, touches the x-axis and turns around B) -1, crosses the x-axis; 3, crosses the x-axis; 1, crosses the x-axis C) 1, crosses the x-axis; -3, crosses the x-axis; -1, touches the x-axis and turns around D) 1, crosses the x-axis; -3, touches the x-axis and turns around; -1, touches the x-axis and turns around 13) f(x) = -x2(x + 8)(x2 – 1) A) 0, touches the x-axis and turns around; -8, crosses the x-axis; -1, crosses the x-axis; 1, crosses the x-axis B) 0, crosses the x-axis; -8, crosses the x-axis; -1, crosses the x-axis; 1, crosses the x-axis C) 0, touches the x-axis and turns around; -8, crosses the x-axis; 1, touches the x-axis and turns around D) 0, touches the x-axis and turns around; 8, crosses the x-axis; -1, touches the x-axis and turns around; 1, touches the x-axis and turns around 14) f(x) = -x2(x + 3)(x2 + 1) A) 0, touches the x-axis and turns around; -3, crosses the x-axis B) 0, touches the x-axis and turns around; 3, crosses the x-axis C) 0, touches the x-axis and turns around; -3, crosses the x-axis; -1, touches the x-axis and turns around D) 0, touches the x-axis and turns around; -3, crosses the x-axis; -1, crosses the x-axis; 1, crosses the x-axis; 15) f(x) = x2(x – 4)(x – 1) A) 0, touches the x-axis and turns around; 4, crosses the x-axis; 1, crosses the x-axis B) 0, touches the x-axis and turns around; -4, crosses the x-axis; -1, crosses the x-axis C) 0, crosses the x-axis; 4, crosses the x-axis; 1, crosses the x-axis D) 0, crosses the x-axis; 4, touches the x-axis and turns around; 1, touches the x-axis and turns around 16) f(x) = -x3(x + 1)2(x – 9) A) 0, crosses the x-axis; -1, touches the x-axis and turns around; 9, crosses the x-axis B) 0, crosses the x-axis; 1, touches the x-axis and turns around; -9, crosses the x-axis C) 0, touches the x-axis and turns around; -1, touches the x-axis and turns around; 9, crosses the x-axis D) 0, touches the x-axis and turns around; 1, crosses the x-axis; 9, crosses the x-axis 17) f(x) = (x – 2)2(x2 – 16) A) 2, touches the x-axis and turns around; -4, crosses the x-axis; 4, crosses the x-axis B) 2, touches the x-axis and turns around; -4, touches the x-axis and turns around; 4, touches the x-axis and turns around C) 2, touches the x-axis and turns around; 16, touches the x-axis and turns around D) -2, touches the x-axis and turns around; 16, crosses the x-axis Page 32 Find the y-intercept of the polynomial function. 18) f(x) = 2x – x3 A) 0 B) 2 C) -1 D) -2 19) f(x) = -x2 + 2x + 3 A) 3 B) -3 C) 0 D) -1 20) f(x) = (x + 1)(x – 8)(x – 1)2 A) -8 B) 8 C) 0 D) -1 21) f(x) = -x2(x + 6)(x2 – 1) A) 0 B) -1 C) -6 D) 6 22) f(x) = -x2(x + 9)(x2 + 1) A) 0 B) 1 C) 9 D) -9 23) f(x) = x2(x – 1)(x – 5) A) 0 B) -5 C) 5 D) -1 24) f(x) = -x2(x + 2)(x – 9) A) 0 B) -9 C) -18 D) 18 25) f(x) = (x – 4)2(x2 – 25) A) -400 B) 400 C) -100 D) 100 Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. 26) f(x) = 6×2 – x3 A) y-axis symmetry B) origin symmetry C) neither 27) f(x) = 8 – x4 A) y-axis symmetry B) origin symmetry C) neither 28) f(x) = x4 – 9×2 A) y-axis symmetry B) origin symmetry C) neither 29) f(x) = x3 – 4x A) origin symmetry B) y-axis symmetry C) neither 30) f(x) = x3 + x2 + 2 A) origin symmetry B) y-axis symmetry C) neither 31) f(x) = x(3 – x2) A) origin symmetry B) y-axis symmetry C) neither 32) x5 – 18×3 + 32x = 0 A) origin symmetry B) y-axis symmetry C) neither 33) f(x) = x3 + 10×2 + 33x + 36 A) origin symmetry B) y-axis symmetry C) neither Page 33 34) f(x) = (x + 1)(x – 8)(x – 1)2 A) y-axis symmetry B) origin symmetry C) neither 35) f(x) = -x2(x + 5)(x2 – 1) A) origin symmetry B) y-axis symmetry C) neither 36) f(x) = -x3(x + 4)2(x – 9) A) origin symmetry B) y-axis symmetry C) neither 37) f(x) = (x – 2)2(x2 – 25) A) origin symmetry B) y-axis symmetry C) neither 38) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 A) y-axis symmetry B) origin symmetry C) neither 39) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 A) origin symmetry B) y-axis symmetry C) neither Page 34 40) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 A) origin symmetry B) y-axis symmetry C) neither 41) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 A) origin symmetry B) y-axis symmetry C) neither Page 35 3 Determine End Behavior MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior to match the function with its graph. 1) f(x) = 2×2 + 2x – 1 A) rises to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 B) falls to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 C) falls to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 D) rises to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 Page 36 2) f(x) = -2×2 – 2x + 1 A) falls to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 B) rises to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 C) rises to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 D) falls to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 Page 37 3) f(x) = 4×3 + 2×2 – 2x + 3 A) falls to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 B) falls to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 C) rises to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 D) rises to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 Page 38 4) f(x) = -8×3 + 3×2 + 4x – 1 A) rises to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 B) rises to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 C) falls to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 D) falls to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 Page 39 5) f(x) = 3×4 – 2×2 A) rises to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 B) falls to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 C) falls to the left and rises to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 D) rises to the left and falls to the right -5 -4 -3 -2 -1 1 2 3 4 5 x y 5 4 3 2 1 -1 -2 -3 -4 -5 Use the Leading Coefficient Test to determine the end behavior of the polynomial function. 6) f(x) = 3×4 – 3×3 – 2×2 + 4x – 2 A) rises to the left and rises to the right B) rises to the left and falls to the right C) falls to the left and rises to the right D) falls to the left and falls to the right 7) f(x) = -4×4 + 2×3 + 2×2 + 5x – 3 A) falls to the left and falls to the right B) rises to the left and falls to the right C) falls to the left and rises to the right D) rises to the left and rises to the right 8) f(x) = 2×3 – 5×2 – 2x + 4 A) falls to the left and rises to the right B) rises to the left and falls to the right C) falls to the left and falls to the right D) rises to the left and rises to the right 9) f(x) = x3 + 2×2 + 5x – 1 A) falls to the left and rises to the right B) rises to the left and falls to the right C) falls to the left and falls to the right D) rises to the left and rises to the right 10) f(x) = -4×3 + 2×2 – 3x + 5 A) rises to the left and falls to the right B) falls to the left and rises to the right C) falls to the left and falls to the right D) rises to the left and rises to the right Page 40 11) f(x) = 3×3 + 5×3 – x5 A) rises to the left and falls to the right B) falls to the left and rises to the right C) falls to the left and falls to the right D) rises to the left and rises to the right 12) f(x) = x – 5×2 – 2×3 A) rises to the left and falls to the right B) falls to the left and rises to the right C) falls to the left and falls to the right D) rises to the left and rises to the right 13) f(x) = (x – 5)(x – 4)(x – 3)2 A) rises to the left and rises to the right B) falls to the left and rises to the right C) rises to the left and falls to the right D) falls to the left and falls to the right 14) f(x) = (x – 5)(x – 4)(x – 2)3 A) falls to the left and rises to the right B) rises to the left and rises to the right C) rises to the left and falls to the right D) falls to the left and falls to the right 15) f(x) = -5(x2 – 1)(x – 3)2 A) falls to the left and falls to the right B) falls to the left and rises to the right C) rises to the left and rises to the right D) rises to the left and falls to the right 16) f(x) = x3(x – 2)(x + 3)2 A) rises to the left and rises to the right B) falls to the left and rises to the right C) rises to the left and falls to the right D) falls to the left and falls to the right 17) f(x) = -x2(x – 2)(x + 3) A) falls to the left and falls to the right B) falls to the left and rises to the right C) rises to the left and falls to the right D) rises to the left and rises to the right 18) f(x) = -6×3(x – 3)(x + 2)2 A) falls to the left and falls to the right B) falls to the left and rises to the right C) rises to the left and falls to the right D) rises to the left and rises to the right Solve the problem. 19) A herd of deer is introduced to a wildlife refuge. The number of deer, N(t), after t years is described by the polynomial function N(t) = -t4 + 25t + 100. Use the Leading Coefficient Test to determine the graph’s end behavior. What does this mean about what will eventually happen to the deer population? A) The deer population in the refuge will die out. B) The deer population in the refuge will grow out of control. C) The deer population in the refuge will reach a constant amount greater than 0. D) The deer population in the refuge will be displaced by “oil” wells. Page 41 20) The following table shows the number of fires in a county for the years , where 1 represents 1994, 2 represents 1995, and so on. Year, x Fires, T 1994, 1 3452 1995, 2 3497.6 1996, 3 3553.38 1997, 4 3597.92 1998, 5 3653.8 This data can be approximated using the third-degree polynomial T(x) = -0.57×3 + 0.51×2 + 62.06x + 3390. Use this function to predict the number of fires in 2005. Round to the nearest whole number. A) 3223 B) 3209 C) -155 D) 2478 21) The following table shows the number of fires in a county for the years , where 1 represents 1994, 2 represents 1995, and so on. Year, x Fires, T 1994, 1 2663.9 1995, 2 2736.04 1996, 3 2771.48 1997, 4 2819.28 1998, 5 2878.5 This data can be approximated using the third-degree polynomial T(x) = -0.49×3 + 0.59×2 + 62.80x + 2601. Use the Leading Coefficient Test to determine the end behavior to the right for the graph of T. Will this function be useful in modeling the number of fires over an extended period of time? Explain your answer. A) The graph of T decreases without bound to the right. This means that as x increases, the values of T will become more and more negative and the function will no longer model the number of fires. B) The graph of T increases without bound to the right. This means that as x increases, the values of T will become large and positive and, since the values of T will become so large, the function will no longer model the number of fires. C) The graph of T approaches zero for large values of x. This means that T will not be useful in modeling the number of fires over an extended period. D) The graph of T decreases without bound to the right. Since the number of larceny thefts will eventually decrease, the function T will be useful in modeling the number of fires over an extended period of time. 4 Use Factoring to Find Zeros of Polynomial Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the zeros of the polynomial function. 1) f(x) = x3 + x2 – 20x A) x = 0, x = – 5, x = 4 B) x = – 5, x = 4 C) x = 3, x = 4 D) x = 0, x = 3, x = 4 Page 42 2) f(x) = x3 + 2×2 – x – 2 A) x = -1, x = 1, x = -2 B) x = 1, x = – 2, x = 2 C) x = – 2, x = 2 D) x = 4 3) f(x) = x3 – 8×2 + 16x A) x = 0, x = 4 B) x = 0, x = -4 C) x = 1, x = 4 D) x = 0, x = -4, x = 4 4) f(x) = x3 + 5×2 – 9x – 45 A) x = -5, x = -3, x = 3 B) x = 5, x = -3, x = 3 C) x = -3, x = 3 D) x = -5, x = 9 5) f(x) = 5(x + 5)(x – 4)2 A) x = -5, x = 4, B) x = 5, x =2 C) x = -5, x =2 D) x = 5, x = -4, x = 2 5 Identify Zeros and Their Multiplicities MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. 1) f(x) = 5(x + 6)(x – 6)4 A) -6, multiplicity 1, crosses x-axis; 6, multiplicity 4, touches x-axis and turns around B) 6, multiplicity 1, crosses x-axis; -6, multiplicity 4, touches x-axis and turns around C) -6, multiplicity 1, touches x-axis and turns around; 6, multiplicity 4, crosses x-axis D) 6, multiplicity 1, touches x-axis and turns around; -6, multiplicity 4, crosses x-axis 2) f(x) = 4(x – 6)(x – 3)3 A) 6, multiplicity 1, crosses x-axis; 3, multiplicity 3, crosses x-axis B) -6, multiplicity 1, crosses x-axis; -3, multiplicity 3, crosses x-axis C) 6, multiplicity 1, crosses x-axis; 3, multiplicity 3, touches x-axis and turns around D) -6, multiplicity 1, touches x-axis; -3, multiplicity 3, touches x-axis and turns around 3) f(x) = -3 x + 4 (x – 2)3 A) – 4, multiplicity 1, crosses x-axis; 2, multiplicity 3, crosses x-axis B) 4, multiplicity 1, crosses x-axis; -2, multiplicity 3, crosses x-axis C) – 4, multiplicity 1, touches the x-axis and turns around; 2, multiplicity 3, touches x-axis and turns around D) 4, multiplicity 1, touches the x-axis and turns around; -2, multiplicity 3, touches x-axis and turns around 4) f(x) = 2(x2 + 4)(x – 2)2 A) 2, multiplicity 2, touches the x-axis and turns around B) -4, multiplicity 1, crosses the x-axis; 2, multiplicity 2, touches the x-axis and turns around. C) -4, multiplicity 1, crosses the x-axis; 2, multiplicity 2, crosses the x-axis D) 2, multiplicity 2, crosses the x-axis Page 43 5) f(x) = 14 x4(x2 – 3)(x + 2) A) 0, multiplicity 4, touches x-axis and turns around; -2, multiplicity 1, crosses x-axis; 3, multiplicity 1, crosses x-axis; – 3, multiplicity 1, crosses x-axis B) 0, multiplicity 4, crosses x-axis; -2, multiplicity 1, touches x-axis and turns around; 3, multiplicity 1, touches x-axis and turns around; – 3, multiplicity 1, touches x-axis and turns around C) 0, multiplicity 4, touches x-axis and turns around; -2, multiplicity 1, crosses x-axis D) 0, multiplicity 4, touches x-axis and turns around; -2, multiplicity 1, crosses x-axis 3, multiplicity 2, touches x-axis and turns around 6) f(x) = x + 15 4 (x + 5)3 A) – 15 , multiplicity 4, touches the x-axis and turns around; -5, multiplicity 3, crosses the x-axis. B) – 15 , multiplicity 4, crosses the x-axis; -5, multiplicity 3, touches the x-axis and turns around C) 15 , multiplicity 4, touches the x-axis and turns around; 5, multiplicity 3, crosses the x-axis. D) 15 , multiplicity 4, crosses the x-axis; 5, multiplicity 3, touches the x-axis and turns around 7) f(x) = x + 12 2 (x2 + 1)5 A) – 12 , multiplicity 2, touches the x-axis and turns around. B) – 12 , multiplicity 2, touches the x-axis and turns around; -1, multiplicity 5, crosses the x-axis C) 12 , multiplicity 2, touches the x-axis and turns around; 1, multiplicity 5, crosses the x-axis D) – 12 , multiplicity 2, crosses the x-axis. Page 44 8) f(x) = x3 + x2 – 12x A) 0, multiplicity 1, crosses the x-axis – 4, multiplicity 1, crosses the x-axis 3, multiplicity 1, crosses the x-axis B) – 4, multiplicity 2, touches the x-axis and turns around 3, multiplicity 1, crosses the x-axis C) 0, multiplicity 1, crosses the x-axis 4, multiplicity 1, crosses the x-axis -3, multiplicity 1, crosses the x-axis D) 0, multiplicity 1, touches the x-axis and turns around; – 4, multiplicity 1, touches the x-axis and turns around; 3, multiplicity 1, touches the x-axis and turns around 9) f(x) = x3 + 8×2 + 20x + 16 A) -2, multiplicity 2, touches the x-axis and turns around; -4, multiplicity 1, crosses the x-axis. B) -2, multiplicity 2, crosses the x-axis; -4, multiplicity 1, touches the x-axis and turns around C) 2, multiplicity 1, crosses the x-axis; -2, multiplicity 1, crosses the x-axis; -4, multiplicity 1, crosses the x-axis. D) 2, multiplicity 1, crosses the x-axis; -2, multiplicity 2, touches the x-axis and turns around; -4, multiplicity 1, crosses the x-axis. 10) f(x) = x3 + 7×2 – x – 7 A) -1, multiplicity 1, crosses the x-axis; 1, multiplicity 1, crosses the x-axis; – 7, multiplicity 1, crosses the x-axis. B) 7, multiplicity 1, crosses the x-axis; 1, multiplicity 1, crosses the x-axis; – 7, multiplicity 1, crosses the x-axis. C) 1, multiplicity 2, touches the x-axis and turns around; – 7, multiplicity 1, crosses the x-axis. D) -1, multiplicity 1, touches the x-axis and turns around; 1, multiplicity 1, touches the x-axis and turns around; – 7, multiplicity 1, touches the x-axis and turns around Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible. 11) Crosses the x-axis at -1, 0, and 4; lies above the x-axis between -1 and 0; lies below the x-axis between 0 and 4. A) f(x) = x3- 3×2 – 4x B) f(x) = x3+ 3×2 – 4x C) f(x) = -x3 + 3×2 + 4x D) f(x) = – x3- 3×2 + 4x 12) Crosses the x-axis at -3, 0, and 1; lies below the x-axis between -3 and 0; lies above the x-axis between 0 and 1. A) f(x) = -x3 – 2×2 + 3x B) f(x) = – x3 + 2×2 + 3x C) f(x) = x3 + 2×2 – 3x D) f(x) = x3 – 2×2 – 3x Page 45 13) Touches the x-axis at 0 and crosses the x-axis at 2; lies below the x-axis between 0 and 2. A) f(x) = x3 – 2×2 B) f(x) = x3 + 2×2 C) f(x) = -x3 + 2×2 D) f(x) = -x3 – 2×2 14) Touches the x-axis at 0 and crosses the x-axis at 2; lies above the x-axis between 0 and 2. A) f(x) = -x3 + 2×2 B) f(x) = x3 + 2×2 C) f(x) = x3 – 2×2 D) f(x) = -x3 – 2×2 6 Use the Intermediate Value Theorem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers. 1) f(x) = 6×3 + 7×2 + 8x + 10; between -2 and -1 A) f(-2) = -26 and f(-1) = 3; yes B) f(-2) = 26 and f(-1) = 3; no C) f(-2) = -26 and f(-1) = -3; no D) f(-2) = 26 and f(-1) = -3; yes 2) f(x) = 4×5 – 9×3 – 6×2 – 6; between 1 and 2 A) f(1) = -17 and f(2) = 26; yes B) f(1) = 17 and f(2) = 26; no C) f(1) = -17 and f(2) = -26; no D) f(1) = 17 and f(2) = -26; yes 3) f(x) = 3×4 – 5×2 – 8; between 1 and 2 A) f(1) = -10 and f(2) = 20; yes B) f(1) = 10 and f(2) = 21; no C) f(1) = -10 and f(2) = -20; no D) f(1) = 10 and f(2) = -20; yes 4) f(x) = -2×4 – 4×3+ 8x + 2; between 1 and 2 A) f(1) = 4 and f(2) = -46; yes B) f(1) = 4 and f(2) = 46; no C) f(1) = -4 and f(2) = -46; no D) f(1) = -4 and f(2) = 46; yes 5) f(x) = 9×3 – 7x – 5; between 1 and 2 A) f(1) = -3 and f(2) = 53; yes B) f(1) = -3 and f(2) = -53; no C) f(1) = 3 and f(2) = 53; no D) f(1) = 3 and f(2) = -53; yes 7 Understand the Relationship Between Degree and Turning Points MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the maximum possible number of turning points for the graph of the function. 1) f(x) = -x2 + 6x + 5 A) 1 B) 2 C) 0 D) 3 2) f(x) = 6×8 – 3×7 – 7x – 23 A) 7 B) 0 C) 6 D) 8 3) f(x) = x6 + 8×7 A) 6 B) 7 C) 8 D) 1 4) g(x) = – 54 x + 1 A) 0 B) 2 C) 1 D) 3 5) f(x) = (x – 7)(x + 7)(6x – 1) A) 2 B) 6 C) 3 D) 0 Page 46 6) f(x) = x2( x2 – 3)(3x – 1) A) 4 B) 5 C) 12 D) 2 7) f(x) = (2x + 5)4( x4 – 5)(x – 3) A) 8 B) 9 C) 18 D) 4 8) f(x) = (x – 5)(x + 3)(x – 3)(x + 1) A) 3 B) 4 C) 0 D) 1 Solve. 9) Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function increasing? A) 0 through 10 and 25 through 40 B) 0 through 40 C) 0 through 10 and 20 through 50 D) 10 through 25 and 40 through 50 10) Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function increasing? A) 0 through 10 and 30 through 50 B) 0 through 50 C) 0 through 20 and 30 through 50 D) 0 through 10 and 40 through 50 Page 47 11) Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function decreasing? A) 10 through 25 and 40 through 50 B) 10 through 50 C) 10 through 25 and 40 through 45 D) 0 through 10 and 25 through 40 12) Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function decreasing? A) 10 through 30 B) 0 through 30 C) 10 through 20 and 30 through 50 D) 0 through 10 and 30 through 50 13) Suppose that a polynomial function is used to model the data shown in the graph below. Determine the degree of the polynomial function of best fit and the sign of the leading coefficient. A) Degree 4; negative leading coefficient. B) Degree 5; positive leading coefficient. C) Degree 5; negative leading coefficient. D) Degree 4; positive leading coefficient. Page 48 14) Suppose that a polynomial function is used to model the data shown in the graph below. Determine the degree of the polynomial function of best fit and the sign of the leading coefficient. A) Degree 3; positive leading coefficient. B) Degree 4; negative leading coefficient. C) Degree 3; negative leading coefficient. D) Degree 4; positive leading coefficient. 15) The profits (in millions) for a company for 8 years were as follows: Year, x Profits, P 1993, 1 1994, 2 1995, 3 1996, 4 1997, 5 1998, 6 1999, 7 2000, 8 1.1 1.7 2.0 1.4 1.3 1.5 1.8 2.1 Which of the following polynomials is the best model for this data? A) P(x) = 0.05×2 – 0.8x + 6 B) P(x) = -0.08×3 + 7×2 + 1.3x – 0.18 C) P(x) = 0.03×3 – 0.3×2 + 1.3x + 0.17 D) P(x) = -0.03×4 – 0.3×2 + 1.3x + 0.17 Page 49 8 Graph Polynomial Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the polynomial function. 1) f(x) = x4 – 4×2 -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) -8 -6 -4 -2 2 4 6 8 x 20 y 16 12 8 4 -4 -8 -12 -16 -20 B) -8 -6 -4 -2 2 4 6 8 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 C) -8 -6 -4 -2 2 4 6 8 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 D) -10 -8 -6 -4 -2 2 4 6 8 10 x 800 y 640 480 320 160 -160 -320 -480 -640 -800 Page 50 2) f(x) = 4×2 – x3 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 51 3) f(x) = 13 – 13 x4 -5 5 x y 5 -5 A) -5 5 x y 5 -5 B) -5 5 x y 5 -5 C) -5 5 x y 5 -5 D) -5 5 x y 5 -5 Page 52 4) f(x) = x3 + 9×2 – x – 9 x y A) -10 -8 -6 -4 -2 2 4 6 8 10 x y 100 80 60 40 20 -20 -40 -60 -80 -100 B) -10 -8 -6 -4 -2 2 4 6 8 10 x y 100 80 60 40 20 -20 -40 -60 -80 -100 C) -10 -8 -6 -4 -2 2 4 6 8 10 x y 500 400 300 200 100 -100 -200 -300 -400 -500 D) -10 -8 -6 -4 -2 2 4 6 8 10 x y 500 400 300 200 100 -100 -200 -300 -400 -500 Page 53 5) f(x) = x3 – 4×2 + x + 6 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 54 6) f(x) = 6x – x3 – x5 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 55 7) f(x) = 6×4 + 9×3 -5 5 x y 5 -5 A) -5 5 x y 5 -5 B) -5 5 x y 5 -5 C) -5 5 x y 5 -5 D) -5 5 x y 5 -5 Page 56 8) f(x) = 6×3 – 5x – x5 -10 -5 5 10 x y 10 5 -5 -10 A) -10 -5 5 10 x y 10 5 -5 -10 B) -10 -5 5 10 x y 10 5 -5 -10 C) -10 -5 5 10 x y 10 5 -5 -10 D) -10 -5 5 10 x y 10 5 -5 -10 Page 57 9) f(x) = x4 – 8×3 + 16×2 x y A) -10 -8 -6 -4 -2 2 4 6 8 10 x y 300 240 180 120 60 -60 -120 -180 -240 -300 B) -10 -8 -6 -4 -2 2 4 6 8 10 x y 300 240 180 120 60 -60 -120 -180 -240 -300 C) -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 x y 1000 800 600 400 200 -200 -400 -600 -800 -1000 D) -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 x y 250 200 150 100 50 -50 -100 -150 -200 -250 Page 58 10) f(x) = x5 – 6×3 – 27x -5 -4 -3 -2 -1 1 2 3 4 5 x 150 y 120 90 60 30 -30 -60 -90 -120 -150 A) -5 -4 -3 -2 -1 1 2 3 4 5 x y 150 120 90 60 30 -30 -60 -90 -120 -150 B) -5 -4 -3 -2 -1 1 2 3 4 5 x y 150 120 90 60 30 -30 -60 -90 -120 -150 C) -5 -4 -3 -2 -1 1 2 3 4 5 x 150 y 120 90 60 30 -30 -60 -90 -120 -150 D) -5 -4 -3 -2 -1 1 2 3 4 5 x 150 y 120 90 60 30 -30 -60 -90 -120 -150 Page 59 11) f(x) = x4 – 2×3 – x2 + 2 -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) -10 -8 -6 -4 -2 2 4 6 8 10 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 B) -10 -8 -6 -4 -2 2 4 6 8 10 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 C) -10 -8 -6 -4 -2 2 4 6 8 10 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 D) -10 -8 -6 -4 -2 2 4 6 8 10 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 Page 60 12) f(x) = x4 + 4×3 + 4×2 -12 -10 -8 -6 -4 -2 2 4 6 8 10 x y 10 8 6 4 2 -2 -4 -6 -8 -10 A) -10 -8 -6 -4 -2 2 4 6 8 10 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 B) -10 -8 -6 -4 -2 2 4 6 8 10 x 10 y 8 6 4 2 -2 -4 -6 -8 -10 C) -10 -8 -6 -4 -2 2 4 6 8 10 x 20 y 16 12 8 4 -4 -8 -12 -16 -20 D) -10 -8 -6 -4 -2 2 4 6 8 10 x 800 y 640 480 320 160 -160 -320 -480 -640 -800 Page 61 13) f(x) = -2x(x + 2)2 -5 5 x y 10 5 -5 -10 A) -5 5 x y 10 5 -5 -10 B) -5 5 x y 10 5 -5 -10 C) -5 5 x y 10 5 -5 -10 D) -5 5 x y 10 5 -5 -10 Page 62 14) f(x) = x(x – 1)(x + 1) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y -1 -2 -3 -4 -5 -6 A) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y -1 -2 -3 -4 -5 -6 B) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y -1 -2 -3 -4 -5 -6 C) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y -1 -2 -3 -4 -5 -6 D) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y -1 -2 -3 -4 -5 -6 Page 63 15) f(x) = -x2(x + 1)(x + 3) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 A) -4 -3 -2 -1 1 2 3 4 x y 20 15 10 5 -5 -10 -15 -20 B) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 C) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 D) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 16) f(x) = (x + 1)2(x2 – 25) x y Page 64 A) -10 -8 -6 -4 -2 2 4 6 8 10 x y 250 200 150 100 50 -50 -100 -150 -200 -250 B) -10 -8 -6 -4 -2 2 4 6 8 10 x y 250 200 150 100 50 -50 -100 -150 -200 -250 C) -10 -8 -6 -4 -2 2 4 6 8 10 x 250 y 200 150 100 50 -50 -100 -150 -200 -250 D) - x y 2500 2000 1500 1000 500 -500 -1000 -1500 -2000 -2500 Page 65 17) f(x) = -x2(x – 1)(x + 1) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 A) -4 -3 -2 -1 1 2 3 4 x y 20 15 10 5 -5 -10 -15 -20 B) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 C) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 D) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 Page 66 18) f(x) = -2×3(x + 1)2(x + 3) x y A) -4 -3 -2 -1 1 2 3 4 x y 160 120 80 40 -40 -80 -120 -160 B) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 C) -4 -3 -2 -1 1 2 3 4 x 20 y 15 10 5 -5 -10 -15 -20 D) -4 -3 -2 -1 1 2 3 4 x 160 y 120 80 40 -40 -80 -120 -160 Page 67 19) f(x) = (x – 5)(x – 3)(x – 2) -6 -4 -2 2 4 6 x y 6 4 2 -2 -4 -6 A) -6 -4 -2 2 4 6 x y 6 4 2 -2 -4 -6 B) -6 -4 -2 2 4 6 x y 6 4 2 -2 -4 -6 C) -6 -4 -2 2 4 6 x y 6 4 2 -2 -4 -6 D) -6 -4 -2 2 4 6 x y 6 4 2 -2 -4 -6 Page 68 20) f(x) = (x + 1)(x + 3)(x + 5)2 -6 -4 -2 2 4 6 x y 12 8 4 -4 -8 -12 A) -6 -4 -2 2 4 6 x y 12 8 4 -4 -8 -12 B) -6 -4 -2 2 4 6 x y 12 8 4 -4 -8 -12 C) -6 -4 -2 2 4 6 x y 12 8 4 -4 -8 -12 D) -6 -4 -2 2 4 6 x y 12 8 4 -4 -8 -12 Page 69 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Complete the following: (a) Use the Leading Coefficient Test to determine the graph’s end behavior. (b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept. (c) Find the y-intercept. (d) Graph the function. 21) f(x) = x2(x + 3) x y 22) f(x) = (x + 2)(x – 3)2 x y Page 70 23) f(x) = -2(x – 1)(x + 3)3 x y 2.4 Dividing Polynomials; Remainder and Factor Theorems 1 Use Long Division to Divide Polynomials MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Divide using long division. 1) (x2 + 13x + 42) ÷ (x + 6) A) x + 7 B) x + 13 C) x2 + 7 D) x2 + 13 2) (8×2 – 67x – 45) ÷ (x – 9) A) 8x + 5 B) 8x – 5 C) x – 67 D) 8×2 + 67 3) (-14×2 + 23x – 3) ÷ (-7x + 1) A) 2x – 3 B) -14x -3 C) x -3 D) -3x + 1 4) 4m3 + 21m2 – 47m + 14 m + 7 A) 4m2 – 7m + 2 B) 4m2 + 7m + 2 C) m2 + 8m + 9 D) m2 + 7m + 4 5) 3r3 – 13r2 – 54r – 14 r – 7 A) 3r2 + 8r + 2 B) 3r2 – 8r – 2 C) 3r2 + 8r + 2 r – 7 D) r2 + 2r + 8 6) (-20×3 + 26×2 – 13x + 4) ÷ (5x – 4) A) -4×2 + 2x – 1 B) -4×2 – 1 C) x2 + 2x – 1 D) x2 – 2x + 1 7) 5×3 – 18x + 4 x + 2 A) 5×2 – 10x + 2 B) 5×2 – 28x + 60 x + 2 C) 5×2 + 28x + 60 x + 2 D) 5×2 + 10x + 2 Page 71 8) (15×3 – 3) ÷ (5x – 1) A) 3×2 + 35 x + 3 25 – 72 25(5x – 1) B) 3×2 + 35 x + 3 25 + 72 25(5x – 1) C) 3×2 + 35 x + 3 25 D) 3×2 – 35 x + 3 25 9) 25×3 + 30×2 – 2x + 9 -5x – 2 A) -5×2 – 4x + 2 + 13 -5x – 2 B) -5×2 – 4x + 2 C) -5×2 – 4x + 2 + 16 -5x – 2 D) x2 + 2 + -4 -5x – 2 10) x4 + 81 x – 3 A) x3 + 3×2 + 9x + 27 + 162 x – 3 B) x3 + 3×2 + 9x + 27 + 81 x – 3 C) x3 + 3×2 + 9x + 27 D) x3 – 3×2 + 9x – 27 + 162 x – 3 11) (5×4 – 3×2 + 15×3 – 9x) ÷ (5x + 15) A) x3 – 35 x B) x3 + 35 x C) x3 – 15x + 25x 5x + 15 D) x3 – 35 x – 18x 5x + 15 12) 8y4 + 12y3 – 2y 2y2 + y A) 4y2 + 4y – 2 B) 4y2 + 8y + 4 + 2y 2y2 + y C) 4y2 + 4y – 6y 2y2 + y D) 4y2 + 6y – 2y 2y2 + y 13) (14×3 + x2 – 42x – 3) ÷ (7×2 – 21) A) 2x + 17 B) 2x +7 C) 2x + -3 7×2 – 21 D) 2x + 3 7×2 – 21 14) (-2×4 + 7×3 + 5×2 – 9x + 20) ÷ (4 – x) A) 2×3 + 1×2 – x + 5 B) 2×3 + 1×2 – x – 5 C) 2×3 + 1×2 – x – 5 + 40 4 – x D) 2×3 + 1×2 + x – 5 15) (-5×5 – x3 + 2×2 + 255x – 14) ÷ (x2 – 7) A) -5×3 – 36x + 2 + 3x x2 – 7 B) -5×3 – 36x + 2 – 3x x2 – 7 C) -5×3 – 36x + 2 + 3x – 28 x2 – 7 D) -5×3 – 36x – 2 + 3x x2 – 7 Page 72 16) x4 – 3×3 – 8×2 + 11x + 9 x2 – 4x – 1 A) x2 + x – 3 + 6 x2 – 4x – 1 B) x2 + x – 3 C) x2 – 8x + 23 + -85x – 26 x2 – 4x – 1 D) x2 – 8x + 23 17) 4t4 + 18t3 – 8t2 – 66t + 36 2t2 + 6t – 4 A) 2t2 + 3t – 9 B) 2t2 – 3t – 9 C) 2t2 + 3t + 9 D) 2t2 + 4t – 9 Solve the problem. 18) A rectangle with width 2x + 1 inches has an area of 2×4 + 7×3 – 17×2 – 58x – 24 square inches. Write a polynomial that represents its length. A) x3 + 3×2 – 10x – 24 inches B) x3 – 10×2 + 3x – 24 inches C) x3 + 5×2 – 9x – 24 inches D) x3 – 9×2 + 5x – 24 inches 19) The width of a rectangle is x – 23 feet and its area is 9×3 + 12×2 + 15x – 18 square feet. Write a polynomial that represents the length of the rectangle. A) 9×2 + 18x + 27 ft B) 9×2 – 18x + 27 ft C) 9×2 + 6x + 11 ft D) 9×2 + 18x – 27 ft 20) Two people are 41 years old and 21 years old, respectively. In x years from now, their ages can be represented by x + 41 and x + 21. Use long division to find the ratio of the older person’s age to the younger person’s age in x years. A) 1 + 20 x + 21 B) 1 + 62 x + 21 C) 1.9524 D) 1 + 62 x + 41 2 Use Synthetic Division to Divide Polynomials MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Divide using synthetic division. 1) (x2 + 17x + 72) ÷ (x + 9) A) x + 8 B) x – 63 C) x2 + 8 D) x3 – 63 2) (x2 + 11x + 27) ÷ (x + 5) A) x + 6 – 3 x + 5 B) x + 6 + 3 x + 5 C) x + 6 x + 5 D) x + 7 3) 5×2 + 33x – 56 x + 8 A) 5x – 7 B) x – 7 C) -7x + 8 D) -5x + 7 4) 5×3 – 17×2 + 7x – 3 x – 3 A) 5×2 – 2x + 1 B) -5×2 + 3x + 1 C) 53 x2 – 17 3 x + 73 D) -5×2 – 3x – 1 Page 73 5) 4×3 + 10×2 + 9x + 10 x + 2 A) 4×2 + 2x + 5 B) -4×2 – 2x + 5 C) 2×2 + 5x + 92 D) 4×2 x + 5 + 5 6) x5 + x3 – 2 x + 3 A) x4 – 3×3 + 10×2 – 30x + 90 + -272 x + 3 B) x4 – 3×3 + 9×2 – 26x + 78 + -236 x + 3 C) x4 – 2×2 + 4 x + 3 D) x4 – 2 + 4 x + 3 7) x4 – 3×3 + x2 + 6x – 8 x – 1 A) x3 – 2×2 – x + 5 – 3 x – 1 B) x3 – 2×2 + x + 7 + 6 x – 1 C) x3 + 2×2 – x + 7 – 3 x – 1 D) x3 – 2×2 + x + 5 + 6 x – 1 8) (x4 + 256) ÷ (x – 4) A) x3 + 4×2 + 16x + 64 + 512 x – 4 B) x3 + 4×2 + 16x + 64 + 256 x – 4 C) x3 + 4×2 + 16x + 64 D) x3 – 4×2 + 16x – 64 + 512 x – 4 9) (x5 – 2×4 – 10×3 + x2 – x + 128) ÷ (x + 3) A) x4 – 5×3 + 5×2 – 14x + 41 + 5 x + 3 B) x4 – 5×3 + 5×2 – 14x – 41 + 5 x + 3 C) x4 – 5×3 + 5×2 – 15x + 41 + 9 x + 3 D) x4 – 5×3 + 5×2 – 15x – 42 + 9 x + 3 10) (5×5 + 6×4 – 4×3 + x2 – x + 31) ÷ (x + 2) A) 5×4 – 4×3 + 4×2 + 7x + 13 + 5 x + 2 B) 5×4 – 4×3 + 4×2 – 7x – 14 + 5 x + 2 C) 5×4 – 4×3 + 4×2 – 8x + 14 + 8 x + 2 D) 5×4 – 4×3 + 4×2 – 8x – 14 + 8 x + 2 3 Evaluate a Polynomial Using the Remainder Theorem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use synthetic division and the Remainder Theorem to find the indicated function value. 1) f(x) = x4 – 4×3 + 2×2 + 9x + 7; f(2) A) 17 B) 34 C) -17 D) 1 2) f(x) = 2×3 – 5×2 – 5x + 8; f(-2) A) -18 B) -14 C) -12 D) -38 Page 74 3) f(x) = 6×4 + 8×3 + 3×2 – 4x + 41; f(3) A) 758 B) 214 C) 1302 D) 2054 4) f(x) = x5 + 5×4 – 8×3 + 3; f(3) A) 435 B) -435 C) 51 D) 192 5) f(x) = x4 – 2×3 – 8×2 + 4x – 8; f 14 A) – 1927 256 B) – 1927 1024 C) 1927 256 D) – 241 32 4 Use the Factor Theorem to Solve a Polynomial Equation MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Use synthetic division to divide f(x) = x3 + 4×2 – 15x – 18 by x + 6. Use the result to find all zeros of f. A) {-6, 3, -1} B) {-6 , -3, 1} C) {6, -3, 1} D) {6, 3, -1} 2) Solve the equation 3×3 – 28×2 + 69x – 20 = 0 given that 4 is a zero of f(x) = 3×3 – 28×2 + 69x – 20. A) 4, 5, 13 B) 4, -5, – 13 C) 4, 1, 53 D) 4, -1, – 53 3) Solve the equation 12×3 – 73×2 + 68x – 15 = 0 given that 13 is a root. A) 13 , 34 , 5 B) 13 , – 34 , -5 C) 13 , 54 , 3 D) 13 , – 54 , -3 Use synthetic division to show that the number given to the right of the equation is a solution of the equation, then solve the polynomial equation. 4) x3 – 5×2 + 2x + 8 = 0; 2 A) {4, -1, 2} B) {-4, -1, 2} C) {4, 1, 2} D) {-4, 1, 2} 5) 2×3 + 10×2 – 4x – 48 = 0; -3 A) 2, -4, -3 B) – 2, -4, -3 C) 2, 4, -3 D) – 2, 4, -3 6) 2×3 – 5×2 – 6x + 9 = 0; 1 A) – 32 , 3, 1 B) 32 , 3, 1 C) – 32 , -3, 1 D) 32 , -3, 1 7) 6×3 + 3×2 – 15x + 6 = 0; 1 A) 12 , -2, 1 B) – 12 , -2, 1 C) 12 , 2, 1 D) – 13 , 3, 1 Page 75 Use the graph or table to determine a solution of the equation. Use synthetic division to verify that this number is a solution o