College Algebra Final Exam questions
with appropriate answers
Remainder Theorem - answersIf polynomial p(x) of degree n>1 is divided by (x-a) where
a is a constant, then the remainder is p(a)
Find minima or maxima - answersfind -b/2a and than plug back in to equation
What strategies can you use to find all the zeros of a polynomial? - answerssynthetic
division, graphing, factoring
Percent Profit Formula - answersPercentage Profit = ((selling price-cost price)/(cost
price))*100
Polynomials - answersMonomial- 5x2
Binomial- 2x2+5x
Trinomial- 5x2+3x+6
The Rational Roots Test p/q - answersThis relationship is always true: If a polynomial
has rational roots, then those roots will be fractions of the form (plus-or-minus) (factor of
the constant term) / (factor of the leading coefficient). However, not all fractions of this
form are necessarily zeroes of the polynomial. Indeed, it may happen that none of the
fractions so formed is actually a zero of the polynomial.
Find all possible rational x-intercepts of x4 + 2x3 - 7x2 - 8x + 12.
The constant term is 12, with factors of 1, 2, 3, 4, 6, and 12. The leading coefficient in
this case is just 1, which makes my work a lot simpler. The Rational Roots Test says
that the possible zeroes are at: Copyright © Elizabeth Stapel 2002-2011 All Rights
Reserved
± 1, 2, 3, 4, 6, 12
= -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12
Graphs of Polynomials: Predicting End Behavior of a Function - answersThe end
behavior of a polynomial function is the behavior of the graph of f(x) as x approaches
positive infinity or negative infinity.
The degree and the leading coefficient of a polynomial function determine the end
behavior of the graph.
, The leading coefficient is significant compared to the other coefficients in the function
for the very large or very small numbers. So, the sign of the leading coefficient is
sufficient to predict the end behavior of the function.
Graphs of Polynomials: Predicting End Behavior of a Function (2) - answersPolynomial
End Behavior:
1. If the degree n of a polynomial is even, then the arms of the graph are either both up
or both down.
2. If the degree n is odd, then one arm of the graph is up and one is down.
3. If the leading coefficient an is positive, the right arm of the graph is up.
4. If the leading coefficient an is negative, the right arm of the graph is down.
Graphs of Polynomials: Predicting End Behavior of a Function (3) - answersTo predict
the end-behavior of a polynomial function, first check whether the function is odd-
degree or even-degree function and whether the leading coefficient is positive or
negative.
Graphs of Polynomials: Turning Points - answersTurning Points
A Turning Point is an x-value where a local maximum or local minimum happens:
How many turning points does a polynomial have? - answersNever more than the
Degree minus 1
How do you determine the degree of a polynomial? - answersThe Degree of a
Polynomial with one variable is the largest exponent of that variable.
How do the graphs of polynomials behave? - answersGraphs will be continuous and
smooth
Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (-
1,1); larger values of n flatten out near 0, and rise more sharply.
Odd exponents behave the same: go from negative x and y to positive x and y; go
through (0,0), (1,1) and (-1,-1); larger values of n flatten out near 0, and fall/rise more
sharply
Factors:
Larger values squash the curve (inwards to y-axis)
Smaller values expand it (away from y-axis)
And negative values flip it upside down
Turning points: there will be "Degree-1" or less.
End Behavior: use the term with the largest exponent
Hooke's Law - answersThe formula for Hooke's Law is "F = kd", where "F" is the force
and "d" is the distance. (Note that, in physics, "weight" is a force. These Hooke's Law
problems are often stated in terms of weight, and the weight is the force.)
with appropriate answers
Remainder Theorem - answersIf polynomial p(x) of degree n>1 is divided by (x-a) where
a is a constant, then the remainder is p(a)
Find minima or maxima - answersfind -b/2a and than plug back in to equation
What strategies can you use to find all the zeros of a polynomial? - answerssynthetic
division, graphing, factoring
Percent Profit Formula - answersPercentage Profit = ((selling price-cost price)/(cost
price))*100
Polynomials - answersMonomial- 5x2
Binomial- 2x2+5x
Trinomial- 5x2+3x+6
The Rational Roots Test p/q - answersThis relationship is always true: If a polynomial
has rational roots, then those roots will be fractions of the form (plus-or-minus) (factor of
the constant term) / (factor of the leading coefficient). However, not all fractions of this
form are necessarily zeroes of the polynomial. Indeed, it may happen that none of the
fractions so formed is actually a zero of the polynomial.
Find all possible rational x-intercepts of x4 + 2x3 - 7x2 - 8x + 12.
The constant term is 12, with factors of 1, 2, 3, 4, 6, and 12. The leading coefficient in
this case is just 1, which makes my work a lot simpler. The Rational Roots Test says
that the possible zeroes are at: Copyright © Elizabeth Stapel 2002-2011 All Rights
Reserved
± 1, 2, 3, 4, 6, 12
= -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12
Graphs of Polynomials: Predicting End Behavior of a Function - answersThe end
behavior of a polynomial function is the behavior of the graph of f(x) as x approaches
positive infinity or negative infinity.
The degree and the leading coefficient of a polynomial function determine the end
behavior of the graph.
, The leading coefficient is significant compared to the other coefficients in the function
for the very large or very small numbers. So, the sign of the leading coefficient is
sufficient to predict the end behavior of the function.
Graphs of Polynomials: Predicting End Behavior of a Function (2) - answersPolynomial
End Behavior:
1. If the degree n of a polynomial is even, then the arms of the graph are either both up
or both down.
2. If the degree n is odd, then one arm of the graph is up and one is down.
3. If the leading coefficient an is positive, the right arm of the graph is up.
4. If the leading coefficient an is negative, the right arm of the graph is down.
Graphs of Polynomials: Predicting End Behavior of a Function (3) - answersTo predict
the end-behavior of a polynomial function, first check whether the function is odd-
degree or even-degree function and whether the leading coefficient is positive or
negative.
Graphs of Polynomials: Turning Points - answersTurning Points
A Turning Point is an x-value where a local maximum or local minimum happens:
How many turning points does a polynomial have? - answersNever more than the
Degree minus 1
How do you determine the degree of a polynomial? - answersThe Degree of a
Polynomial with one variable is the largest exponent of that variable.
How do the graphs of polynomials behave? - answersGraphs will be continuous and
smooth
Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (-
1,1); larger values of n flatten out near 0, and rise more sharply.
Odd exponents behave the same: go from negative x and y to positive x and y; go
through (0,0), (1,1) and (-1,-1); larger values of n flatten out near 0, and fall/rise more
sharply
Factors:
Larger values squash the curve (inwards to y-axis)
Smaller values expand it (away from y-axis)
And negative values flip it upside down
Turning points: there will be "Degree-1" or less.
End Behavior: use the term with the largest exponent
Hooke's Law - answersThe formula for Hooke's Law is "F = kd", where "F" is the force
and "d" is the distance. (Note that, in physics, "weight" is a force. These Hooke's Law
problems are often stated in terms of weight, and the weight is the force.)
