2015/16TI-Smartview

2.5 The Fundamental

Theorem of Algebra

HWQ

2 52 2

ii i

Perform the operation and simplify. Work must be shown to receive

credit.

2.5 The Fundamental Theorem of Algebra

Students will find all zeros of polynomial functions, including complex zeros.

Students will find conjugate pairs of complex zeros.

Students will find zeros of polynomials by factoring.

The Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, then f(x) has n zeros, real and (or) imaginary.

Example 1: Real Zeros of a Polynomial Function

Counting multiplicity, justify that the second-degree polynomial function has exactly two factors and zeros.

f x x x( ) 2 6 9

Example 2: Real and Imaginary Complex Zeros of a Polynomial Function

Justify that the third degree polynomial functionhas exactly three factors and zeros.

f x x x( ) 3 4

Example 3: Finding the zeros of a Polynomial Function

Write as the product of linear factors, and list all the zeros of f.

f x x x x x( ) 5 3 22 12 8

Example 4: Finding a Polynomial with Given Zeros

Find a fourth degree polynomial function with real coefficients that has - 1, - 1, and 3i as zeros.

• Find all zeros without graphing the polynomial first given is a zero:

1 3i

4 3 23 6 2 60x x x x

Example 4: More practice

Example 5: Factoring a PolynomialFor the following polynomial list all zeros.

f x x x( ) 4 2 20

You try:

• Solve by factoring: 4 2 2 0x x

• Solve: 4 3 23 5 21 22 0x x x x

Example 6: More practice

Homework

2.5 pg. 140 #9-35 odd, 45, 47, 57

**QUIZ TOMORROW OVER SECTIONS 2.4 & 2.5**