2.5 The Fundamental Theorem of Algebra

1. 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.

2. 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.

3. 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.

4. Example 1: Real Zeros of a Polynomial Function Counting multiplicity, justify that the second-degree polynomial funcion has exactly two factors and zeros.

5. Example 2: Real and Imaginary Complex Zeros of a Polynomial Function Justify that the third degree polynomial function has exactly three factors and zeros.

6. y 2 x –2 Example 3: Finding the zeros of a Polynomial Function Write as the product of linear factors, and list all the zeros of f.

7. 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.

8. Example 5: Factoring a Polynomial Write the polynomial: • as a product of quadratic factors. • as a product of linear factors. • in complete factored form.

9. y 2 x –2 Example 5: Write as a product of quadratic factors, linear factors, in complete factored form

10. y 2 x –2 Example 6: Finding the zeros of a polynomial function Find all the zeros of given that 1 + 3i is a zero of f.