MHF4U1-ASSIGNMENT CHAPTER 2 A NAME:___________________________
True/False
Indicate whether the statement is true or false.
____ 1. When performing long division of a polynomial by a linear binomial, the degree of the remainder is always
smaller than the degree of the divisor.
____ 2. If P(–3) = 0 for a polynomial P(x), then x + 3 is a factor of P(x).
____ 3. For a polynomial equation P(x) = 0, if P(x) is not factorable, then P(x) = 0 has no real roots.
____ 4. All quartic polynomial equations have at least one real solution.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 5. If x3 – 4x2 + 5x – 6 is divided by x – 1, then the restriction on x is
a. x –4 c. x 1
b. x –1 d. no restrictions
____ 6. What is the remainder when x4 + 2x2 – 3x + 7 is divided by x + 2?
a. 25 c. 37
b. 13 d. 9
____ 7. If 6x4 – 2x3 – 21x2 + 7x + 8 is divided by 3x – 1 to give a quotient of 2x3 – 7x and
a remainder of 8, then which of the following is true?
a.
b. 6x4 – 2x3 – 21x2 + 7x + 8 = (3x – 1)(2x3 – 7x) + 8
c.
d. all of the above
____ 8. When P(x) = 4x3 – 4x + 1 is divided by 2x – 3, the remainder is
a. c.
b. P(3) = 97 d.
____ 9. For a polynomial P(x), if P = 0, then which of the following must be a factor of P(x)?
a. c. 5x + 3
b. 3x + 5 d. 5x – 3
____ 10. Which of the following binomials is a factor of x3 – 6x2 + 11x – 6?
a. x – 1 c. x + 7
b. x + 1 d. 2x + 3
____ 11. Which set of values for x should be tested to determine the possible zeros of x3 – 2x2 + 3x – 12?
, a. 1, 2, 3, 4, 6, and 12 c. ±1, ±2, ±3, ±4, and ±6
b. ±1, ±2, ±3, ±4, ±6, and ±12 d. ±2, ±3, ±4, ±6, and ±12
____ 12. Determine the value of k so that x – 3 is a factor of x3 – 3x2 + x + k.
a. k = 3 c. k = 1
b. k = –3 d. k = –1
____ 13. Find k if 2x + 1 is a factor of kx3 + 7x2 + kx – 3.
a. k = –2 c.
k=
b. k = 2 d. none of the above
____ 14. Which of the following is the fully factored form of x3 + 3x2 – x – 3?
a. (x + 3)(x2 – 1) c. x2(x + 3) – (x + 3)
b. (x – 1)(x + 1)(x + 3) d. (x2 – 1)(x – 3)
____ 15. Which of the following is the fully factored form of x3 – 6x2 – 6x – 7?
a. (x – 7)(x + 1)2 c. (x – 7)(x2 + x + 1)
b. (x – 7)(x + 1)(x – 1) d. (x – 6)(x + 1)(x – 1)
____ 16. Which of the following is the factored form of x4 – 2x2 – 3?
a. (x – 1)(x + 1)(x – 3) c. (x2 + 1)(x2 – 3)
b. (x – 1)(x + 3)
2
d. none of the above
____ 17. One root of the equation x3 + 2x – 3x2 – 6 = 0 is
a. –3 c. 3
b. –1 d. 1
____ 18. What is the maximum number of real distinct roots that a quartic equation can have?
a. infinitely many c. 2
b. 4 d. none of the above
____ 19. If 2 is one root of the equation 4x3 + kx – 24 = 0, then the value of k is
a. –1 c. 8
b. –4 d. impossible to determine
____ 20. Based on the graph of f(x) = x4 – 2x3 + 3x + 2 shown, what are the real roots of x4 – 2x3 + 3x + 2 = 0?
a. 2 c. impossible to determine
b. –2, –1, 1, 2 d. no real roots
True/False
Indicate whether the statement is true or false.
____ 1. When performing long division of a polynomial by a linear binomial, the degree of the remainder is always
smaller than the degree of the divisor.
____ 2. If P(–3) = 0 for a polynomial P(x), then x + 3 is a factor of P(x).
____ 3. For a polynomial equation P(x) = 0, if P(x) is not factorable, then P(x) = 0 has no real roots.
____ 4. All quartic polynomial equations have at least one real solution.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 5. If x3 – 4x2 + 5x – 6 is divided by x – 1, then the restriction on x is
a. x –4 c. x 1
b. x –1 d. no restrictions
____ 6. What is the remainder when x4 + 2x2 – 3x + 7 is divided by x + 2?
a. 25 c. 37
b. 13 d. 9
____ 7. If 6x4 – 2x3 – 21x2 + 7x + 8 is divided by 3x – 1 to give a quotient of 2x3 – 7x and
a remainder of 8, then which of the following is true?
a.
b. 6x4 – 2x3 – 21x2 + 7x + 8 = (3x – 1)(2x3 – 7x) + 8
c.
d. all of the above
____ 8. When P(x) = 4x3 – 4x + 1 is divided by 2x – 3, the remainder is
a. c.
b. P(3) = 97 d.
____ 9. For a polynomial P(x), if P = 0, then which of the following must be a factor of P(x)?
a. c. 5x + 3
b. 3x + 5 d. 5x – 3
____ 10. Which of the following binomials is a factor of x3 – 6x2 + 11x – 6?
a. x – 1 c. x + 7
b. x + 1 d. 2x + 3
____ 11. Which set of values for x should be tested to determine the possible zeros of x3 – 2x2 + 3x – 12?
, a. 1, 2, 3, 4, 6, and 12 c. ±1, ±2, ±3, ±4, and ±6
b. ±1, ±2, ±3, ±4, ±6, and ±12 d. ±2, ±3, ±4, ±6, and ±12
____ 12. Determine the value of k so that x – 3 is a factor of x3 – 3x2 + x + k.
a. k = 3 c. k = 1
b. k = –3 d. k = –1
____ 13. Find k if 2x + 1 is a factor of kx3 + 7x2 + kx – 3.
a. k = –2 c.
k=
b. k = 2 d. none of the above
____ 14. Which of the following is the fully factored form of x3 + 3x2 – x – 3?
a. (x + 3)(x2 – 1) c. x2(x + 3) – (x + 3)
b. (x – 1)(x + 1)(x + 3) d. (x2 – 1)(x – 3)
____ 15. Which of the following is the fully factored form of x3 – 6x2 – 6x – 7?
a. (x – 7)(x + 1)2 c. (x – 7)(x2 + x + 1)
b. (x – 7)(x + 1)(x – 1) d. (x – 6)(x + 1)(x – 1)
____ 16. Which of the following is the factored form of x4 – 2x2 – 3?
a. (x – 1)(x + 1)(x – 3) c. (x2 + 1)(x2 – 3)
b. (x – 1)(x + 3)
2
d. none of the above
____ 17. One root of the equation x3 + 2x – 3x2 – 6 = 0 is
a. –3 c. 3
b. –1 d. 1
____ 18. What is the maximum number of real distinct roots that a quartic equation can have?
a. infinitely many c. 2
b. 4 d. none of the above
____ 19. If 2 is one root of the equation 4x3 + kx – 24 = 0, then the value of k is
a. –1 c. 8
b. –4 d. impossible to determine
____ 20. Based on the graph of f(x) = x4 – 2x3 + 3x + 2 shown, what are the real roots of x4 – 2x3 + 3x + 2 = 0?
a. 2 c. impossible to determine
b. –2, –1, 1, 2 d. no real roots
