Lesson 7 2 Finding Complex Solutions of Polynomial

Lesson 7. 2: Finding Complex Solutions of Polynomial Equations The multiplicity of a zero/root is the number of times that the related factor occurs in the factorization. Ex. p(x) = (x – 2)2(x + 3)3(x – 1) zeros are 2 (mult. 2), -3 (mult. 3), and 1

Examples Find all zeros (including non-real zeros) and include any multiplicities greater than 1. 1) p(x) = x 3 + 7 x 2 2) p(x) = x 3 – 64 3) p(x) = x 4 – 16 4) p(x) = x 4 + 5 x 3 + 6 x 2 – 4 x – 8

The Fundamental Theorem of Algebra and Its Corollary Fundamental Theorem of Algebra Every polynomial of degree n ≥ 1 has at least one zero, where a zero may be a complex number. Corollary of Fundamental Theorem of Algebra Every polynomial of degree n ≥ 1 has exactly n zeros, including multiplicities.

Example Find all roots of x 4 – 6 x 2 – 27 = 0

Irrational Root Theorem If a polynomial p(x) has rational coefficients and is a root of the equation p(x) = 0, where a and b are rational and is irrational, then is also a root of p(x) = 0. In other words, irrational roots come in pairs. Ex. If is a root, so is .

Complex Conjugates Theorem If is a non-real root of a polynomial equation with real-number coefficients, then is also a root. In other words, non-real roots come in complex conjugate pairs. Ex. If 3 + 2 i is a root, so is 3 – 2 i.

Example Write a function of least degree and leading coefficient of 1 with the following roots and put in standard form. Roots: 3, 2 – 5 i

Example Write a function of least degree and leading coefficient of 1 with the following roots and put in standard form. Roots: 5,