Mth 111 – College Algebra Test Review for Polynomials and Rational Functions
Polynomial Functions
1.
2.
3.
4.
5.
6.
Recognize whether a function is a polynomial or not.
Determine the degree and leading coefficient of P(x).
For any given value c, determine P(c) using synthetic division.
For any given value c, determine if c is a zero of P(x) using synthetic division.
For any given value c, determine if (x – c) is a linear factor of P(x) using synthetic division.
For any zero of P(x), rewrite P(x) as the product of a one linear factor and one non-linear factor using
synthetic division.
7. Understand the connections between these four statements:
 P(c) = 0;
 (x – c) is a binomial factor of P(x);
 C is a zero of P(x);
 (x, 0) is an x-intercept of P(x)
8. Given a polynomial, determine the number of possible bends, real zeros and end behavior.
9. Given the graph of a polynomial, determine: degree, real zeros, and multiplicity of each zero, end
behavior, x-axis behavior, x-intercepts, and y-intercepts.
10. Given a polynomial in factored form, determine: degree, real zeros, multiplicity of each zero, end
behavior, x-axis behavior, x-intercepts, and y-intercepts.
11. Write a polynomial in factored form given real zeros and their multiplicity.
12. Write a polynomial in factored form given the graph and degree of P(x) and one additional point.
13. Write a third degree polynomial in the form: P( x )  ax 3  bx 2  cx  d , given real zeros and
their multiplicity.
14. Write the third degree P(x) in the form: P( x )  ax 3  bx 2  cx  d , given the zeros - one real and
15.
16.
17.
18.
one complex.
Given a third degree polynomial with one real zero, write P(x) as the product of three linear factors
– one real and two complex.
Given the graph of P(x), evaluate P(c) for any given values of c on the graph.
Given the graph of P(x), solve equations in the form P(c)=0 for c.
Given the graph of P(x) and the value of “d”, solve equations in the form P(c)=d for c.
Rational Functions
1. Given a rational function determine all the horizontal and vertical asymptotes, if any.
2. Graph a rational function indicating the asymptotes with dashed lines.
Mth 111 – College Algebra Test Review for Polynomials and Rational Functions
(1) Recognize whether a function is a polynomial or not.
y  x 2  2
y  x 3  2x 2  x  2
y  2x  x  3
y  x3  2
(2) Determine the degree and leading coefficient of P(x).
P( x )  4x 3  2x 4  x 2  x  1
P( x)  3(2x  3)( x  2)(4x  1)
P( x)  x(3x  1)2
(3-5) Given P( x)  x3  4 x 2  x  6 , show how to use synthetic division to:
(3) Determine P(4).
(4) Determine if 2 is a zero of P(x).
(5) Determine if (x – 3) is a binomial factor of P(x).
(3-5) Given P( x)  x 4  13x 2  36 , show how to use synthetic division to:
(3) Determine P(-5).
(4) Determine if -3 is a zero of P(x).
(5) Determine if (x – 2) is a binomial factor of P(x).
(6) Given ( x  3) is a factor of P( x )  x 3  x 2  x  15 rewrite P(x) as the product of a binomial
factor and its reduced polynomial Q( x ) using synthetic division.
(7) Given that “-5 is a zero of P(x)” , make three other equivalent statements related to this fact.
(8) For each polynomial, determine the number of possible bends, possible real zeros and end behavior.
P( x )  x 4  bx 2  c
P( x )   x 3  bx 2  cx  d
P( x )   x 6  bx 3  cx  d
(9) Given the graph of the polynomial determine the
degree, real zeros, and multiplicity of each zero.
Write P(x) in factored form with leading coefficient
equal to 1. What is the y-intercept of P(x)?
(10) Given P( x )  ( x  2)2 ( x  1)( x  3) determine the degree, real zeros, multiplicity of each zero,
end behavior, x-axis behavior (cross or bounce), x-intercepts, and y-intercepts.
(11) Write any fourth degree polynomial in factored form
given that 2, -5 and 0 are zeros of P(x) with 0 having
multiplicity 2.
(12) Using the graph, write a fifth degree polynomial in
factored form passing through (-3, -16).
(12) Write a third degree polynomial with zeros: 0, 3, -5 and
passing through the point: (2, 28)
(13) Write a third degree polynomial in the form: P( x )  ax 3  bx 2  cx  d with real zeros 2 and -3
with 2 having multiplicity 2.
(14) Write the third degree P(x) in the form: P( x )  ax 3  bx 2  cx  d , given the zeros 4 and 3i
(15) Given P( x)  x3  5x 2  16 x  80 , write P(x) as the product of three binomial factors.
(15) Given P( x)  x 4  16 x 2  225 , find all complex zeros of P(x).
(16) skip
y=4
(17-19) The graph of P(x) to the right as a scale of “1” for
both the x- and y-axis.
a)
b)
c)
d)
Evaluate P(-3.5)
Solve for c, P(c)=0.
P(x)
Solve for c, P(c)=4
Find the exact third degree polynomial that matches the graph.
Bonus: What it is all about!
Solve the polynomial equation: x 4  25  26 x2
Rational Functions:
1.
Given f ( x ) 
3x
determine all horizontal and vertical asymptotes, if any.
x2  1
2. Graph the rational function indicating the asymptotes with dashed lines.