Question

How do you factor completely `5x^4 +10x^2 -15`?

Answer 1

`5x^4+10x^2-15`

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

`=5(x-sqrt(3)i)(x+sqrt(3)i)(x-1)(x+1)`

Explanation:

• Separate out the common scalar factor `5`.

• Factor as a quadratic in `x^2`.

• Use the difference of squares identity:

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

as follows:

`5x^4+10x^2-15`

`=5(x^4+2x^2-3)`

`=5((x^2)^2+2(x^2)-3)`

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

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

Then if we allow Complex coefficients...

`=5(x^2-(sqrt(3)i)^2)(x-1)(x+1)`

`=5(x-sqrt(3)i)(x+sqrt(3)i)(x-1)(x+1)`

Answer 2

`5(x^2 + 3)(x + 1)(x - 1)`

Explanation:

Divide the common factor of 5 out first.

This is a disguised quadratic: `5((x^2)^2 + 2(x)^2 - 3)`

Find factors of 3 which subtract to give 2.

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

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