Question

How do you factor `x^3-8x^2+19x-12`?

Answer 1

• We have the polynomial `color(red)(p(x) = x^3-8x^2+19x-12`
  As the degree of the polynomial is greater than 2, we need to use Synthetic Division to factorize it.

• In Synthetic Division, we first find a value of `x` that makes this polynomial equal to zero. That is, we need to find the ZERO of the polynomial. We do that by Trial and Error.
  For `x=color(blue)1`, this Polynomial will equal zero.

`p(1) = 1^3-8*1^2+19*1-12 = 1 - 8 + 19 -12 = 0`

As `x=1` makes this polynomial zero, we know for sure that `color(green)((x-1))` is one of the factors of the polynomial.

• To find the other two factors, we use the process of Synthetic Division. To explain the entire process of Synthetic Division using text is a bit difficult, but this image should help:

The 1, -7 and 12 give us another factor of the polynomial:
`x^2 - 7x + 12`

We have reduced p(x) to

`p(x) = (x-1)(x^2-7x+12)`

But `x^2-7x+12` can be factorised further
`=x^2-3x-4x+12`
`=x(x-3)-4(x-3)`
`=(x-3)(x-4)`

So finally, we get

`p(x) = (x-1)(x-3)(x-4)`

`x^3-8x^2+19x-12 = color(green)( (x-1)(x-3)(x-4)`