Download 2.7 Apply the Fundamental Theorem of Algebra

2.7
Apply the Fundamental
Theorem of Algebra
Polynomials Quiz: Tomorrow
(over factoring and Long/Synthetic Division)
Polynomials Test: Friday
Vocabulary
When a factor of a polynomial appears
more than once, you can count its solution
more than once. The repeated factor
produces a repeated solution.
 If f(x) is a polynomial of degree n where
n > 0, then the equation f(x) = 0 has
exactly n solutions provided each solution
repeated twice is counted as 2 solutions,
each solution repeated three times is
counted as 3 solutions, and so on.

Vocabulary

Complex Conjugates Theorem:

If f is a polynomial function with real
coefficients, and a + bi is an imaginary zero of
f, then a – bi is also a zero of f.
Vocabulary:

Irrational Conjugates Theorem:

Suppose f is a polynomial function with
rational coefficients, and a and b are rational
numbers such that b is irrational. If a  b
is a zero of f, then a  b is also a zero of f.
Example 1:

Find all zeros of
f(x) = x5 – x4 – 7x3 + 11x2 + 16x – 20
Step 1: Find all the rational zeros of f.
-2 is a zero repeated twice and 1 is a zero, so…
(x + 2)(x + 2)(x – 1)
**(x + 2)(x + 2)(x – 1) does not multiply out to
give us the original function**

Example 1: Continued

-2
Step 2: Divide f by its known factors x + 2,
x + 2, and x - 1
1 -1 -7 11 16 - 20
Now, divide what you got by
the next factor. And so on
until you have divided with all
factors

Step 3: Find the complex zeros.
You Try:

Find all the zeros of the function
f(x) = x4 – 4x3 + 5x2 – 2x - 12
Example 2:

Write a polynomial function of least
degree that has rational coefficients, a
leading coefficient of 1, and 4 and 1 2
as zeros.
You Try:

Write a polynomial function f of least
degree that has rational coefficients, a
leading coefficient of 1, and the given
zeros.
 i, 4
Homework:

Study for Polynomial Quiz tomorrow


I will post the answers to the Factoring WS
online.
Look over Long/Synthetic Division