Question

How do you factor `x^4-8x^2+16`?

Answer 1

`(x^2-4)^2`

Explanation:

Make a dumby variable, in this case I will use `u`.
Let `u=x^2`
You have `x^4-8x^2+16`.
This is equivalent to `(x^2)^2-8x^2+16`.
You can now plug in `u` wherever `x^2` is.
You get `u^2-8u+16`.
Now, you can factor this like a normal polynomial.
`(u-4)(u-4)`
or
`(u-4)^2`.
Finally, you plug in `u` back into the expression.
The factored form will be `(x^2-4)^2`.

Answer 2

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

Explanation:

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

Answer 3

Notice how this is almost a simple quadratic trinomial. In fact, it can be thought of as a quadratic of `x^2`.

Let `color(blue)(u = x^2)`

`x^4 - 8x^2 + 16`

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

`= color(blue)u^2 - 8color(blue)u + 16`

We can easily factor this quadratic.

`= (color(blue)u - 4)^2`

`= (color(blue)(x^2) - 4)^2`