2.5 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra – If f(x) is a polynomial of degree n, where.

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Presentation transcript:

2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra 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 Linear Factorization Theorem – If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors f(x) = a n (x –c 1 )(x-c 2 )…(x-c n ) Where c 1, c 2,…c n are complex numbers

Find the zeros of the polynomial f(x) = x 3 + 4x f(x) = x 2 - 6x + 9 f(x) = x 5 + x 3 + 2x 2 – 12x + 8

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

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

Finding the Zeros Find all the zeros of f(x) = x 4 -3x 3 + 6x 2 + 2x – 60 given that 1 + 3i is a zero

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. “irreducible over the reals” A quadratic with no real zeros is “irreducible over the reals”

Factoring a Polynomial Write the polynomial f(x) = x 4 – x 2 – 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

Homework Page 140 #’s ODD