Polynomials Conjugate Zeros Theorem If a polynomial has

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

Polynomials: Conjugate Zeros Theorem 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). difference sum Next, the last two factors represent the product of a difference and a sum. Therefore, these two factors multiply as (x – 3)2 – i 2. This simplifies to, x 2 – 6 x + 9 – (- 1) or x 2 – 6 x + 10. Table of Contents Slide 2

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

Polynomials: Conjugate Zeros Theorem Table of Contents