Multiply the factors and write in standard form . 1) 2) 3)

Review – Complex Numbers Name: _________________ Multiply the factors and write in standard form. 1) 2) 3) Solve each equation in the complex number system: 4) 5) 6)

I. Complex Zeros 1. How many zeros does this function have?

Complex Zeros 2. How many zeros does this function have?

An example with no real zeros 3. How many zeros does this function have? How can you be sure there are no real zeros when you graph it on the calculator?

Section 3.7 Complex Zeros Complex Zeros Factoring with Complex Zeros Conjugate Pairs Theorem Finding Complex Zeros

I. Complex Zeros Fundamental Theorem of Algebra • A polynomial of degree n will have exactly • nzeros • and • factors in the complex plane

II. Factoring with real zeros Find the zeros and write as a product of linear factors. Example: All real and rational zeros Find rational zeros on calculator first.

II. Factoring with complex zeros Find the zeros and write as a product of linear factors. Example: No real zeros (all pure imaginary zeros)

Example: real and complex zeros. Factor completely Calculator shows rational zeros are at: 4 and -7

Example: real, irrational, complex zeros Calculator shows rational zeros are at: 4 and -3 Find rational zeros first.

III. Conjugate Pairs Theorem If a + bi is a zero of f, then a – bi is also a zero of f Determine the zeros Complex zeros always come in conjugate pairs!

III. a) Corollary to Conjugate Pairs Theorem Suppose a function has degree and zeros as given, what are the remaining zeros? A polynomial f of odd degree with real coefficients has at least one real zero. • Degree 3; zeros: 1, 2 + i • Degree 6; zeros 3 + 2i, i, -4 + i Can a polynomial of degree 3 have as zeros 2i, 4-i ? Will a polynomial of degree 4 have real zeros if it has complex zeros 4-i, and 5i ?

IV. Linear Factors of Complex Zeros Form a polynomial with real coefficients satisfying : degree 4 and zeros at: 3, multiplicity 2; and 2) degree 3 and zeros at: 4, and

V. Given a complex zero, find remaining Complex Zeros Suppose the function has as a zero. Determine the remaining zeros of the function. Determine the matching pair of complex zeros. Multiply linear factors of the complex zeros. Polynomial division. Solve for zeros of q(x)

VI. Find Complex Zeros and write in factored form Suppose the function has as a zero. Write in factored form:

VI. Find Complex Zeros and write in factored form (p. 237 #29). Suppose the function has as a zero. Determine the remaining zeros of the function Write in factored form

Multiply the factors and write in standard form . 1) 2) 3)