Question

How do you factor `y= x^5 - 16x^3 + 8x^2 - 128` ?

Answer 1

Factor by grouping then using a couple of identities to find:

`x^5-16x^3+8x^2-128 = (x+2)(x^2-2x+4)(x-4)(x+4)`

Explanation:

The sum of cubes identity can be written:

`a^3+b^3=(a+b)(a^2-ab+b^2)`

The difference of squares identity can be written:

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

Factor by grouping then use these two identities as follows:

`x^5-16x^3+8x^2-128`

`=(x^5-16x^3)+(8x^2-128)`

`=x^3(x^2-16)+8(x^2-16)`

`=(x^3+8)(x^2-16)`

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

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

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