Question

How do you simplify `(x^4-256)/(x-4)`?

Answer 1

Hi,

I propose another answer.

1) First, you use this famous formula :

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

You can prove that if you expand the second member.

Take now `y=4`. Because `4^4=256`, you get :

`x^4-256 = (x-4)(x^3+4x^2 + 16x+64)`.

If `x\ne 4`, you can divide by `x-4` and

`\frac{x^4-256}{x-4} = x^3+4x^2+16x+64`.

2) If you know complex numbers, you can have a better factorization.

The equation `x^4=256` has 4 solutions in

`\mathbb{C} : 4,4i,-4i,-4`

Then, you can write, for all

`x\in \mathbb{C}`,

`x^4-256 = (x-4)(x-4i)(x+4i)(x+4)`

and `\frac{x^4-256}{x-4} =(x+4)(x-4i)(x+4i)`.

If you want a real factorization, write

`(x-4i)(x+4i) = (x^2+16)`.

Conclusion `\frac{x^4-256}{x-4} =(x+4)(x^2+16)`.

3) If you don't know complex numbers, no stress!

Remark that

`x^3+4x^2+16x+64` has a easy root : `x=-4`

then `x^3+4x^2+16x+64 = (x+4)(ax^2+bx+c)`

Develop and find that `a=1`, `b=0` and `c=16`.

You find again

`\frac{x^4-256}{x-4} =(x+4)(x^2+16)`.