Topic: Theorems About Roots of Polynomial Equations
Today, we will focus our attention on theorems that will help us solve polynomial equations.
Standards/Goals:
 N.CN.7.: I can solve quadratic equations with real coefficients that have complex solutions.
 I can use the Rational Root Theorem to solve equations.
 I can use the Conjugate Root Theorem to solve equations.
 I can use the Descartes’ Rule of Signs to determine the number of roots of a polynomial equation.
 I can use synthetic division to divide two polynomials.
We want to first look at the Rational Root Theorem
Rational Root Theorem
This theorem says that any rational zero of a function must be in the form of 𝑝 𝑞
,
 where ‘p’ is a factor of the constant term and
 ‘q’ is a factor of the leading coefficient.
This theorem will NOT help us find irrational or imaginary zeros; to find these we must use the reduced polynomial and quadratic formula.
EXAMPLES: List all of the potential rational zeros of the following polynomials. Then use polynomial division and the quadratic formula to identify the actual zeros.
1.
f(x) = 3𝑥
3
+ 5𝑥
2
− 26𝑥 + 8
2.
p(x) = 𝑥 4 − 5𝑥 3 + 10𝑥 2 − 20𝑥 + 24
We also want to familiarize ourselves with the CONJUGATE ROOTS THEOREM.
Conjugate Roots Theorem
This theorem basically says that when we consider a function that is a polynomial with only real coefficients, if a + bi is a zero of ‘f’, then so is the complex number a – bi. In terms of the linear factors of ‘f’, this means that x – (a + bi) is a factor of f, then so is x – (a – bi).
EXAMPLES:
#1. A quartic polynomial P(x) has rational coefficients. If 2 – i and −√5 are roots of P(x) = 0, what are the other roots?
#2. A cubic polynomial P(x) has rational coefficients. If 2 + 3i and 2/3 are two roots of P(x) = 0, what is one additional root?
1

#3. What is a quartic polynomial function P(x) with rational coefficients so that P(x) = 0 has roots 1,
3, and 1 + 2i?
#4. A cubic polynomial P(x) with rational coefficients so that P(x) = 0 has roots -3 and i?
EXAMPLES: Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to factor the following polynomial equations completely, making use of the given zero).
#1. f(x) = 𝑥 4 + 3𝑥 3 + 21𝑥 2 + 147𝑥 − 1372 ; -7i is a zero
#2. p(x) = 𝑥
4
− 2𝑥
3
+ 14𝑥
2
− 8𝑥 + 40 ; 2i is a zero.
We will also use the Descartes’ Rule of Signs to determine how many roots there are in a polynomial.
Descartes’ Rule of Signs
Let P(x) be a polynomial with real coefficients written in standard form.
 The number of positive real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less than that by an even number.
 The number of negative real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(-x) or is less than that by an even number.
In both cases, count multiple roots according to their multiplicity.
EXAMPLES:
#1. What does Descartes’ Rule of Signs tell you about the real roots of 𝑥 3
#2. What does Descartes Rule of Signs tell you about the real roots of −𝑥 4
0 ?
− 𝑥 2 + 𝑥 + 1 = 0 ?
+ 𝑥 3 − 2𝑥 2 + 𝑥 + 1 =
2

Use the Rational Root Theorem to list all possible rational roots for each equation.
1. x 3 + 5x
2. 36x 3
2  2x

15 = 0
+ 144x 2

x

4 = 0
Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots.
3. 2x 3 + 5x 2
4. 12x 4
+ 4x + 1 = 0
+ 14x 3 
5x 2 
14x

4 = 0
A polynomial function P(x) with rational coefficients has the given roots. Find two additional roots of P(x) = 0.
5. 2 + 3i and 7
6. 3

2 and 1
7.

4i and 6

i

3
Write a polynomial function with rational coefficients so that P(x) = 0 has the given roots.
8. 4 and 6
9.  5 and  1
10. 3i and 6
11. 2 + i and 1

5
12.

5 and 3i
EXAMPLES: Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to factor the following polynomial equations completely, making use of the given zero).
13. 𝑓(𝑥) = 𝑥 4
14. 𝑓(𝑥) = 𝑥 4
− 5𝑥
+ 7𝑥
3
3
+ 19𝑥
+ 46𝑥
2
2
− 125𝑥 − 150 ; -5i is a zero.
+ 252𝑥 + 360; −6𝑖 𝑖𝑠 𝑎 𝑧𝑒𝑟𝑜.
What does Descartes’ Rule of Signs say about the number of positive real roots and negative real roots for each polynomial function?
15. P(x) = 3x
16. P(x) = 2x 4
17. P(x) = 4x 5
3 + x 2 
8x

12

x 3 
3x + 7

x 4 
x 3 + 6x 2 
5
3