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Example: Find a third degree polynomial that has the following zeros: 1, 3 – i. Write the polynomial in terms with descending powers of x. Polynomials: Conjugate Zeros Theorem. If a polynomial has nonreal zeros, they occur in conjugate pairs.
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Example: Find a third degree polynomial that has the following zeros: 1, 3 – i. Write the polynomial in terms with descending powers of x. 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. 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)).
difference sum 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). Next, the last two factors represent the product of a difference and a sum. Therefore, these two factors multiply as (x – 3)2 – i2. This simplifies to, x2– 6x + 9 – (- 1) or x2– 6x + 10. Slide 2
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. Polynomials: Conjugate Zeros Theorem Therefore, P(x) = (x – 1)(x2– 6x + 10). Multiplying the binomial and trinomial results in: P(x) = x3– 7x2+ 16x – 10. Note, this is not the only third degree polynomial with zeros, 1, 3 + i, and 3 – i. For example, f (x) = 2x3– 14x2+ 32x – 20 has the same zeros since f (x) is just 2P(x). P(x) = x4– 3x3+ 9x2+ 13x. Slide 3
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