Question

Find all other zeros of P(x) = x? – 8x + 32, given that 2=2i is a zero.If there is more than one zero, separate them with commas.)

Answer 1

Given:

`f(x)=x^3-8x+32`

The zero is (2-2i).

Required:

We need to find the zeros of the given function f(x).

Step-by-step explanation:

The zeros of the function f(x) is 2-2i.

Use the synthetic method.

The given f(x) can be written as follows.

`x^3-8x+32=(x-(2-2i))(x^2+(2-2i)x+(-8+(2-2i)^2))`

`=(x-(2-2i))(x^2+(2-2i)x+(-8+4-8i-4))`

`=(x-(2-2i))(x^2+(2-2i)x+(-8-8i))`
`Use\text{ \lparen2-2i\rparen x=\lparen-2-2i\rparen x+4x}`

`=(x-(2-2i))(x^2+(-2-2i)x+4x+(-8-8i))`

`=(x-(2-2i))(x(x+(-2-2i))+4(x+(-2-2i)))`

`=(x-(2-2i))(x+(-2-2i))(x+4)`

`=(x-(2-2i))(x-(2+2i))(x+4)`

`=(x-(2-2i))(x-(2+2i))(x-4)`

The zeros of the function f(x) are (2-2i), (2+2i), and (-4).

Final answer:

`(2+2i),-4.`