Question

How do you factor `x^3 - 9x^2 + 24x - 20`?

Answer 1

`x^3-9x^2+24x-20 = (x-2)(x-2)(x-5)`

Explanation:

`f(x) = x^3-9x^2+24x-20`

By the rational root theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `20` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1, +-2, +-4, +-5, +-10, +-20`

Trying each in turn, we find:

`f(2) = 8-9(4)+24(2)-20 = 8-36+48-20 = 0`

So `x=2` is a zero and `(x-2)` a factor:

`x^3-9x^2+24x-20 = (x-2)(x^2-7x+10)`

Note that `7 = 2+5` and `10 = 2*5`

So we find:

`x^2-7x+10 = (x-2)(x-5)`

Putting it all together:

`x^3-9x^2+24x-20 = (x-2)(x-2)(x-5)`