Question

How do you factor `u^4-81`?

Answer 1

`u^4-81 = (u-3)(u+3)(u^2+9)`

Explanation:

The difference of squares identity can be written:

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

We can use this a couple of times to derive the factors with Real coefficients, as follows:

`u^4-81 = (u^2)^2-9^2`

`color(white)(u^4-81) = (u^2-9)(u^2+9)`

`color(white)(u^4-81) = (u^2-3^2)(u^2+9)`

`color(white)(u^4-81) = (u-3)(u+3)(u^2+9)`

The remaining quadratic factor has no simpler linear factors with Real coefficients since `u^2+9 >= 9` for any Real value of `u`, hence no Real zeros or corresponding linear factors.