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

1 Answer
mason m
Jun 23, 2016

(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.

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