1. Table of Contents Polynomials: Conjugate Zeros Theorem If a polynomial has nonreal zeros, they occur in conjugate pairs. For example, if 5 + 2i is a zero of a polynomial, 5 – 2i is also a zero. A third degree polynomial has 3 zeros and only 2 are given. Example: Find a third degree polynomial that has the following zeros: 1, 3 – i. Write the polynomial in terms with descending powers of x. However since 3 – i is a zero, 3 + i is also a zero. The polynomial has 3 variable factors, each of the form, x – zero. Therefore a third degree polynomial is: P(x) = (x – 1)(x – (3 + i))(x – (3 – i)).

2. Table of Contents Polynomials: Conjugate Zeros Theorem Slide 2 P(x) = (x – 1)(x – (3 + i))(x – (3 – i)). Next, to write the polynomial in terms with descending powers of x, first distribute to remove the innermost parentheses. P(x) = (x – 1)(x – 3 – i)(x – 3 + i). Next, the last two factors represent the product of a difference and a sum. difference sum Therefore, these two factors multiply as (x – 3) 2 – i 2. This simplifies to, x 2 – 6x + 9 – (- 1) or x 2 – 6x + 10.

3. Table of Contents Polynomials: Conjugate Zeros Theorem Slide 3 Therefore, P(x) = (x – 1)(x 2 – 6x + 10). Multiplying the binomial and trinomial results in: P(x) = x 3 – 7x 2 + 16x – 10. Note, this is not the only third degree polynomial with zeros, 1, 3 + i, and 3 – i. For example, f (x) = 2x 3 – 14x 2 + 32x – 20 has the same zeros since f (x) is just 2P(x). Try:Find a fourth degree polynomial that has the following zeros: 0, - 1, 2 + 3i. Write the polynomial in terms with descending powers of x. P(x) = x 4 – 3x 3 + 9x 2 + 13x.

4. Table of Contents Polynomials: Conjugate Zeros Theorem