Question

How do you find all zeros of `f(x)=5x^4+15x^2+10`?

Answer 1

`f(x)` has zeros `+-i` and `+-sqrt(2)i`

Explanation:

We will use the difference of squares identity, which can be written:

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

with `a=x` and `b=i` or `b=sqrt(2)i` as follows:

`f(x) = 5x^4+15x^2+10`

`color(white)(f(x)) = 5(x^4+3x^2+2)`

`color(white)(f(x)) = 5(x^2+1)(x^2+2)`

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

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

Hence the zeros of `f(x)` are:

`x = +-i`

`x = +-sqrt(2)i`