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

3 Answers
Mar 13, 2018

(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.

Mar 13, 2018

(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

Mar 13, 2018

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