Question

How do you find all zeros with multiplicities of `f(x)=3x^4-14x^2-5`?

Answer 1

`x = +-sqrt(3)/3i`

`x = +-sqrt(5)`

Explanation:

`f(x) = 3x^4-14x^2-5`

`color(white)(f(x)) = (3x^4-15x^2)+(x^2-5)`

`color(white)(f(x)) = 3x^2(x^2-5)+1(x^2-5)`

`color(white)(f(x)) = (3x^2+1)(x^2-5)`

`color(white)(f(x)) = ((sqrt(3)x)^2-i^2)(x^2-(sqrt(5))^2)`

`color(white)(f(x)) = (sqrt(3)x-i)(sqrt(3)x+i)(x-sqrt(5))(x+sqrt(5))`

Hence zeros:

`x = +-i/sqrt(3) = +-sqrt(3)/3i`

`x = +-sqrt(5)`