Question

How do you factor `4x^3 - x^2 -12x + 3` by grouping?

Answer 1

`(4x-1)(x^2-3)`

Explanation:

Think of this cubic as two groups:

`overbrace((4x^3-x^2))^"Group 1"+overbrace((-12x+3))^"Group 2"`

We will want to find a common factor in each group. From `4x^3-x^2`, we see a common factor of `x^2`:

`overbrace(x^2(4x-1))^"Group 1"+overbrace((-12x+3))^"Group 2"`

From Group 2, we could either factor out a `+3` or a `-3`. To make the leading term positive, we will factor out a `-3` from both `-12x` and `3`, leaving:

`overbrace(x^2(4x-1))^"Group 1"+overbrace(-3(4x-1))^"Group 2"`

From here, notice that there is a common factor between Group 1 and Group 2. Both have terms, `x^2` and `-3`, which are being multiplied by the other term `(4x-1)`. Here, `(4x-1)`, despite it being made up of two terms, is the common factor and can be factored out. The `x^2` and `-3` are combined since they share this common factor:

`(4x-1)(x^2-3)`

Depending on your level of instruction, you may recognize that `x^2-3` is a difference of squares, and can be factorized yet. If not, this is a fine final answer.