Question

How do you factor `x^3 – 4x^2 +x + 6`?

Answer 1

`(x-2)(x-3)(x+1)`

Explanation:

It is usually really, really hard to factorize a cubic function. However, for this polynomial, we can factor by grouping. We try values for splitting the term `-4x^2`.

For example, we split it into `-2x^2-2x^2`.

The equation becomes this: `(x^3-2x^2)-(2x^2-x-6)`. We can factorize each of the expressions in the parentheses: `x^2(x-2)-(x-2)(2x+3)`. There is a common factor `(x-2)`.

Factoring the common factor out, we get `(x-2)(x^2-2x-3)`. We then factorize `x^2-2x-3` to `(x-3)(x+1)`.

The fully factored form is then `(x-2)(x-3)(x+1)`.

Answer 2

`x^3-4x^2+x+6=color(magenta)((x-2)(x-3)(x+1))`

Explanation:

Provided the expression has rational roots, we can use the rational root theorem.

For the expression `color(green)(x^3-4x^2+x+6)`
according to the rational root theorem, possible rational roots are:
`color(white)("XXX"){+-1,+-2,+-3,+-6}`

With the use of a spreadsheet these values can be easily checked (it can also be done with a calculator or even manually with a bit more effort).

From this, we see that `x=2`, `x=3`, and `x=-1` all are zeros for the given expression.
This implies that `(x-2)`, `(x-3)`, and `(x+1)` are factors of the given expression.

Since `x^3-4x^2+x+6` is of degree `3`, it has a maximum of `3` factors and we have found them all.