1. Theorems about Roots of Polynomial Equations and
The fundamental theorem of Algebra
What you’ll learn
To solve equations using the Rational Root Theorem.
To use the conjugate Root Theorem.
To use the Fundamental Theorem of Algebra to
solve polynomial equations with complex solutions
Vocabulary
Rational Root Theorem.
Conjugate Root Theorem.
Descartes’ Rule of Signs.
Fundamental theorem of Algebra.

2. Factoring a polynomial can be challenging, but there is a theorem to help you with that.
Constant term,
Example:
Factors of the constant term:
Factors of the leading coefficient:

3. Problem 1: Finding a Rational Root
What are the rational roots of
Leading coefficient 2 (±1,±2) and constant term 5 (±1, ±5)
The only possible rational
roots have the form
The only possible roots are x 1 -1 5 -5
1/2
-1/2
5/2
-5/2
P(x)
Note: the Rational Root Theorem does not necessarily give the
zeros of the equation. It provides a list of first guesses to
test as roots

4. What are the rational roots of
Your turn
What are the rational roots of
Leading coefficient 1 (±1) and constant term 10 (±1, ±2,±5±10)
The only possible rational
roots have the form x 1 -1 2 -2 5 -5 10
-10
P(x)

5. Problem 2: Using the Rational Root Theorem
Once you find one root, use synthetic division to factor the polynomial. Continue finding roots and dividing until you have a second degree polynomial. Use the Quadratic Formula to find the remaining roots
Problem 2: Using the Rational Root Theorem
Leading coefficient 15 (±1,±3,±5,±15) and constant term 2(±1, ±2)
The only possible rational roots have the form
Test each possible rational root in
until you find a root
Test:

6. So 2 is a root, now factor the polynomial using synthetic division

7. Your turn: Answer: 2,-1,-3/2 Remember steps to find rational roots.
Get the constant term factors and the leading
coefficient of the polynomial.
2. Find all possible rational roots
3.Test each possible rational root until you find a root.
4. Factor the polynomial until you get a quadratic.
(you can use synthetic division, quadratic formula,
a calculator, or any other method).when using
TI-84 or 85 store the polynomial in using the
Y= menu. Store the root to be tested in x(enter the
number then press the key STO).
Use VARS Y-VARS 1: Function to evaluate.
Your turn:
Answer: 2,-1,-3/2

8. Conjugate Root Theorem: If P(x) is a polynomial with
Did you remember what is a conjugate number?
If a complex number or an irrational number is a
root of a polynomial equation with rational coefficients,
so is its conjugate.
Conjugate Root Theorem: If P(x) is a polynomial with
rational coefficients, then the irrational roots of P(x)=0
that have a form occur in conjugate pairs. That is,
If is an irrational root with rational then
is also a root.
If P(x) is a polynomial with real coefficients, then the
complex roots of P(x)=0 occur in conjugate pairs. That
is is a complex root with real, then

9. Problem 3:Using the Conjugate Root Theorem to Identify Roots.
Answers:
Remember all rational numbers are real numbers.
Your turn
Answer

10. Problem 4: Using the Conjugates to construct A Polynomial
Answer:
Answer:

11. Descartes’ Rule of Signs Theorem: Let P(x) be a polynomial
with real coefficients written in standard form.
*The number of positive real roots of P(x) 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
This theorem implies that two tests must be done on a
polynomial function to determine both positive and negative
real roots.

12. Problem 5: Using the Descartes’ Rule of Signs.
There are two sign changes, + to – and - to +.
Therefore, there are either 0 or 2 positive real roots.
To find the negative real root you need to plug in -x
into the polynomial
There is only one negative real root
Graph the function and recall that cubic functions have
zero or two turning points. Because the graph already
shows two turning points, it will not change the direction
again. So there are no positive real roots.

13. Your turn Take a note: the degree of a polynomial equation tells
Answers: there are 3 or 1 positive real roots and 1 negative
root. The graph confirm one negative and one positive real
Root
Real roots can be confirmed graphically because they are
x-intercepts. Complex roots cannot be confirmed graphically
because they have an imaginary component.
Take a note: the degree of a polynomial equation tells
you how many roots the equation has. That is the
result of the Fundamental theorem of Algebra provided by
the German mathematician Carl Friedrich Gauss
( )

14. The fundamental Theorem of Algebra: If P(x) is a
polynomial of degree n≥1, then P(x)=0 has exactly
n roots, including multiple and complex roots
Problem 6: Using the Fundamental Theorem of Algebra
I need to find the zeros(5) and using the rational
root and factor theorem, synthetic division and factoring

15. (x-1)(x-2)(x+2)(x-i)(x+i) so the roots are 1,-2,2,i,-i
The factors are -4 and 1
Answers:
(x-1)(x-2)(x+2)(x-i)(x+i) so the roots are 1,-2,2,i,-i
Your turn
Since there is no constant term, make the equation
equal to zero and factor x from the polynomial
Answers: 0,1,-5,2

16. Problem 7: Finding all the zeros of a Polynomial Function
Step 1: use the graphing calculator to find any real roots
Step 2: Factor out the linear factors
(x-3)(x+3).Use Synthetic division twice.
Step3: use the quadratic formula to find the complex roots
Step 4:

17. Your turn
Answer:
The Fundamental Theorem of Algebra: Here are
equivalent ways to state the fundamental theorem
of Algebra. You can use any of these statements to
prove the others.
*Every polynomial equation of degree n≥1 has exactly
n roots, including multiple and complex roots.
*Every polynomial of degree n≥1 has n factors.
*Every polynomial function of degree n≥1 has at least
one complex zero.

18. Classwork odd Homework even
TB pg 316 exercises 9-41 and pg322 exercises 8-37