Algebra 2
Finding Rational Solutions of Polynomial Equations
Date: ___________
EXPLORE. Relating Zeros and Coefficients of Polynomial Functions
The zeros of a polynomial function and the coefficients of the function are related. We are
going to use the two functions in the table below to discover their relationship.
Factored Form
𝑓(𝑥) = (𝑥 + 2)(𝑥 − 1)(𝑥 + 3)
𝑔(𝑥) = (2𝑥 + 3)(4𝑥 − 5)(6𝑥 − 1)
Zeros
Standard Form
Look for a relationship between the zeros of each function and the coefficients of its standard form
function.
Reflect.
1. In general, how are the zeros of a polynomial function related to its standard form function?
Rational Zero Theorem
𝑚
𝑚
If the polynomial 𝑝(𝑥) is a polynomial function with integer coefficients, and if 𝑛 is a zero of 𝑝(𝑥) (𝑝 ( 𝑛 ) = 0),
then 𝑚 is a factor of the constant term of 𝑝(𝑥) and 𝑛 is a factor of the leading coefficient of 𝑝(𝑥).
Rational Root Theorem
If the polynomial 𝑝(𝑥) has integer coefficients, then every rational root of the
𝑚
polynomial equation 𝑝(𝑥) = 0 can be written in the form 𝑛 , where 𝑚 is a factor
of the constant term of 𝑝(𝑥) and 𝑛 is a factor of the leading coefficient of 𝑝(𝑥).
Find the rational zeros of the polynomial function; then write the function as a product of
factors. Make sure to test the possible zeros to find the actual zeros of the function.
A. 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 19𝑥 − 20
B. 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 − 7𝑥 2 + 22𝑥 + 24
C. 𝑝(𝑥) = 2𝑥 4 + 𝑥 3 − 19𝑥 2 − 9𝑥 + 9
Finding Zeros Using the Rational Zero Theorem
Learning Target A: I can use the Rational Zero Theorem to find the rational zeros of a polynomial
function.
If a polynomial function p(x) is equal to (𝑎1 𝑥 + 𝑏1 )(𝑎2 𝑥 + 𝑏2 )(𝑎3 𝑥 + 𝑏3 ), where
𝑎1 , 𝑎2 , 𝑎3 , 𝑏1 , 𝑏2 , 𝑎𝑛𝑑 𝑏3 are integers, the leading coefficient of p(x) will be the product 𝑎1 𝑎2 𝑎3 and
𝑏
𝑏
𝑏
the constant term will be the product 𝑏1 𝑏2 𝑏3. The zeros will be the rational numbers − 𝑎1 , − 𝑎2 , − 𝑎3 .
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Comparing the zeros of p(x) to its coefficient and constant term shows that the numerators of the
polynomial’s zeros are factors of the ___________________ ____________ and the denominators of the zeros
are factors of the ______________________ _________________________________.
Rational Zero Theorem
𝑚
𝑚
If the polynomial 𝑝(𝑥) is a polynomial function with integer coefficients, and if 𝑛 is a zero of 𝑝(𝑥) (𝑝 ( 𝑛 ) = 0),
then 𝑚 is a factor of the constant term of 𝑝(𝑥) and 𝑛 is a factor of the leading coefficient of 𝑝(𝑥).
Rational Root Theorem
If the polynomial 𝑝(𝑥) has integer coefficients, then every rational root of the
𝑚
polynomial equation 𝑝(𝑥) = 0 can be written in the form , where 𝑚 is a factor
𝑛
of the constant term of 𝑝(𝑥) and 𝑛 is a factor of the leading coefficient of 𝑝(𝑥).
Find the rational zeros of the polynomial function; then write the function as a product of
factors. Make sure to test the possible zeros to find the actual zeros of the function.
A. 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 19𝑥 − 20
B. 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 − 7𝑥 2 + 22𝑥 + 24
C. 𝑝(𝑥) = 2𝑥 4 + 𝑥 3 − 19𝑥 2 − 9𝑥 + 9
H. Algebra 2
7.1 Notes
B. 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 − 7𝑥 2 + 22𝑥 + 24
C. 𝑝(𝑥) = 2𝑥 4 + 𝑥 3 − 19𝑥 2 − 9𝑥 + 9
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H. Algebra 2
7.1 Notes
Solving a Real-World Problem Using the Rational Root Theorem
Learning Target B: I can apply the Rational Root Theorem to solve real-world problems modeled by
polynomial functions.
Since a zero of a function f(x) is the value of x for which 𝑓(𝑥) = 0, finding the zeros of a polynomial
function p(x) is the same as finding the _______________________ of the polynomial equation 𝑝(𝑥) = 0.
Because a solution of a polynomial equation is known as a root, the Rational Zero Theorem can be
also expressed as the Rational _______________ Theorem.
Rational Root Theorem
If the polynomial 𝑝(𝑥) has integer coefficients, then every rational root of the
𝑚
polynomial equation 𝑝(𝑥) = 0 can be written in the form 𝑛 , where 𝑚 is a factor
of the constant term of 𝑝(𝑥) and 𝑛 is a factor of the leading coefficient of 𝑝(𝑥).
Engineering A pen company is designing a gift container for their new premium pen. The
marketing department has designed a pyramidal box with a rectangular base. The base width is 1
inch shorter than its base length and the height is 3 inches taller than 3 times the base length. The
volume of the box must be 6 cubic inches. What are the dimensions of the box? Graph the volume
function and the line y = 6 on a graphing calculator to check your solution.
Engineering A box company is designing a new rectangular gift container. The marketing
department has designed a box with a width 2 inches shorter than its length and a height 3 inches
taller than its length. The volume of the box must be 56 cubic inches. What are the dimensions of
the box?
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