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).

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:

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