1. 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

2. Examples
Find all zeros (including non-real zeros) and include any multiplicities greater than 1.
p(x) = x3 + 7x2
p(x) = x3 – 64
p(x) = x4 – 16
p(x) = x4 + 5x3 + 6x2 – 4x – 8

3. 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.

4. Example
Find all roots of x4 – 6x2 – 27 = 0

5. 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 .

6. 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 + 2i is a root, so is 3 – 2i.

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

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