f(x) has a zero of 1 and a zero of -2 with multiplicity of 2. Use this information to find the rest of the zeroes

Question

f(x) has a zero of 1 and a zero of -2 with multiplicity of 2. Use this information to find the rest of the zeroes

f (x) = x⁵ + 3x⁴ + 9x³ + 23x² - 36

The remaining two imaginary solutions are ___________ and ___________.

Answer 1

So, f(x) has factors (x - 1) and (x + 2)². We dividing using Synthetic division:

1 I 1 3 9 23 0 -36

1 4 13 36 36

-2 I 1 4 13 36 36 0

-2 -4 -18 -36

-2 I 1 2 9 18 0

-2 0 -18

1 0 9 0

We get quotient x² + 9 = 0; x = ±2i

Answer 2

The polynomial f(x) has a zero of 1. This means that f(x) can be divided by (x-1) with the remainder 0.

The polynomial f(x) has a zero of -2 with multiplicity of 2. This means that f(x) can be divided by (x+2)² with the remainder 0.

Let divide f(x) by (x-1)(x+2)² = x³ + 3x² - 4.

(x⁵ + 3x⁴ + 9x³ + 23x² - 36) ÷ (x³ + 3x² - 4) = x² + 9

So, the remaining zeros are solutions of the equation x² + 9 = 0, i.e. ±3i.