4.6: Complex Zeros

1. HW: Worksheet 4.6: Complex Zeros You will be able find a polynomial function with specified zeroes and factor polynomials completely.

2. Theorem • A polynomial of n degree has n complex zeroes. • Complex numbers include both real and imaginary numbers • Take the form a + bi. • Complex zeroes occur in conjugate pairs.

3. Example 1 • Find a 3rd degree function such some zeroes are 2, 6 – i. F(x) = (x - 2) (x – (6 – i)) (x – (6 + i)) = (x - 2) ((x – 6) + i) ((x – 6) - i)) = (x - 2) ((x – 6)2 - i2) = (x - 2) ( x2 – 12x + 36 + 1) = (x - 2) ( x2 – 12x + 37) = x3 – 12x2 + 37x - 2x2 + 24x – 74 F(x) =x3 – 14x2 + 61x - 74

4. Example 2 • Find a 3rd degree function such some zeroes are -2, 4 + i.

5. Factoring Completely. • Find the complex zeroes of each polynomial function. • Write f(x) in factored form.

6. Factor, given a zero… h(t) = t3 + 3t2 + 25t + 75 zero = -5i (t2 + 25) is a factor of h(t). t2 + 25 t3 + 3t2 + 25t + 75 3t 2 + 75 + 3 t h(t) = (t + 3)(h + 5i)(h – 5i) - t3 - 25t -3t2 - 75

7. Try This… Factor m(x) = 3x4 + 5x3 + 25x2 + 45x – 18. zero: 3i

8. Factor, given no zeroes… k(x) = x3 + 13x2 + 57x + 85 PRZ: -85, -17, -5, -1 F(-1) = 40 F(-5) = 0 -5 1 13 57 85 -5 -40 -85 1 8 17 0 x2 + 8x + 17 (x2 + 8x + 16) + 1 (x + 4)2 – (i)2

9. Continued … x = {-5, -4 + i, -4 – i} (x + 4)2 – (i)2 ((x + 4) – i)((x + 4) + i) (x+ (4 – i))(x+ (4 + i)) (x– (-4 + i))(x– (-4 – i)) h(x) = (x + 5) [x – (-4 + i)][x – (-4 – i)]

10. Try This… Factor: p(x) = x4 + 6x3 + 11x2 + 12x + 18