Question

How do you factor `z^4 - 10z^2 + 9`?

Answer 1

`z^4-10z^2+9 = (z-1)(z+1)(z-3)(z+3)`

Explanation:

The difference of square identity can be written:

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

We will use this a couple of times.

Let us treat this as a quadratic in `z^2`. Note that the sum of the coefficients is `0`. That is `1-10+9 = 0`. Hence `((z^2)-1)` is a factor and the other factor must be `((z^2)-9)`...

`z^4-10z^2+9 = (z^2)^2-10(z^2)-9`

`color(white)(z^4-10z^2+9) = (z^2-1)(z^2-9)`

`color(white)(z^4-10z^2+9) = (z^2-1^2)(z^2-3^2)`

`color(white)(z^4-10z^2+9) = (z-1)(z+1)(z-3)(z+3)`