2.5 The Fundamental Theorem of Algebra

Students will use the fundamental theorem of algebra to determine the number of zeros of a polynomial.

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 has at least one zero in the complex number system.

* Remember that the polynomial has at most n zeros.

Linear Factorization Theorem

If f(x) is a polynomial of degree n, f has precisely n linear factors.

Where are zeros that are complex numbers.

f x a x z x z x zn n( ) ( )( )...( ) 1 2

z z z1 2 3, ,

Example 1: Real Zeros of a Polynomial Function

Counting multiplicity, justify that the second-degree polynomial funcionhas 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 8y

x–2

2

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.

Example 5: Factoring a Polynomial

Write the polynomial:

a) as a product of quadratic factors.

b) as a product of linear factors.

c) in complete factored form.

f x x x( ) 4 2 20

Example 5:Write as a product of quadratic factors, linear factors, in complete

factored form

y

x–2

2

f x x x( ) 4 2 20

Example 6: Finding the zeros of a polynomial function

Find all the zeros of given that 1 + 3i is a zero of f.

f x x x x x( ) 4 3 23 6 2 60

y

x–2

2