Question

For the following function, one zero is given. Find all the others

f(x) = x3 - 10x2 + 33x - 34; 4 - i
The zeros of f(x) are 4 - i, , and I
(Simplify your answers. Type any nonreal complex zeros first.

Answer 1

4 - i, 4 + i, 2.

Explanation:

Complex zeros exist as conjugate pairs so another zero is 4 + i.

So one factor of f(x) is

(x - (4 + i))(x - (4 - i))

= (x - 4 - i)(x - 4 + i)

= x^2 - 4x + ix - 4x + 16 - 4i - ix + 4i - i^2

= x^2 - 8x + 16 - (-1)

= x^2 - 8x + 17

Dividing:

x^2 - 8x + 17 ) x3 - 10x2 + 33x - 34( x - 2

x3 - 8x^2 + 17x

-2x^2 + 16x - 34

-2x^2 + 16x - 34

So the third root is 2:

x - 2 = 0

x = 2.