Question

How do you find all zeros of `f(x)=2x^4-2x^2-40`?

Answer 1

`x = +-sqrt(5)" "` or `" "x = +-2i`

Explanation:

Given:

`f(x) = 2x^4-2x^2-40`

We can first treat this as a quadratic in `x^2` then factor the two resulting quadratic factors using the difference of squares identity:

`a^2-b^2 = (a-b)(a+b)`

as follows:

`f(x) = 2x^4-2x^2-40`

`color(white)(f(x)) = 2((x^2)^2-x^2-20)`

`color(white)(f(x)) = 2(x^2-5)(x^2+4)`

`color(white)(f(x)) = 2(x^2-(sqrt(5))^2)(x^2-(2i)^2)`

`color(white)(f(x)) = 2(x-sqrt(5))(x+sqrt(5))(x-2i)(x+2i)`

Hence zeros:

`x = +-sqrt(5)" "` or `" "x = +-2i`