2.5 The Fundamental T heorem of Algebra

1. 2.5 The Fundamental Theorem of Algebra

2. The Fundamental Theorem of Algebra • If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system • Linear Factorization Theorem • If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors f(x) = an (x –c1)(x-c2)…(x-cn) Where c1, c2,…cn are complex numbers

3. Find the zeros of the polynomial f(x) = x3 + 4x f(x) = x2 - 6x + 9 f(x) = x5 + x3 + 2x2 – 12x + 8

4. Complex Zeros Occur in Conjugate Pairs Let f(x) be a polynomial function that has real coefficients. If a + bi, where b ≠ 0, is a zero of the function, then the complex a – bi is also a zero of the function. Therefore: if 3 + 2i is a zero, then 3 – 2i is a zero

5. Writing a Polynomial with Given Zeros Zeros: -1, 2, 3i Zeros: 0, 3, 5, 2 – 5i

6. Finding the Zeros Find all the zeros of f(x) = x4 -3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero

7. Factoring a Polynomial • Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factor have no real zeros. • A quadratic with no real zeros is “irreducible over the reals”

8. Factoring a Polynomial Write the polynomial f(x) = x4 – x2 – 20 a) as the product of factors that are irreducible over the rationals b) as the product of factors that are irreducible over the reals c) in completely factored form

9. Homework Page 140 #’s 27-49 ODD