Factoring by Grouping
We often see the grouping method applied to polynomials with 4 terms. The idea is to pair like terms together so that we can apply the distributive property in order to factorize them nicely.
Factor \( x^3 - 3x^2 -x + 3 \).
We have
\[ \begin{align} x^3 - 3x^2 - x + 3 &= x^2(x - 3) - (x - 3) \\ &= (x^2 -1)(x - 3) \\ &= (x - 1)(x + 1)(x - 3). \ _\square \end{align} \]
Factor \( x^2 + 4x - y^2 - 4y \).
We have
\[ \begin{align} x^2 + 4x - y^2 - 4y &= (x^2 - y^2) + (4x - 4y) \\ &= (x -y)(x + y) + 4(x-y) \\ &= (x - y)(x + y + 4). \ _\square \end{align} \]
Factor \( x^2 - 16y^2 - 6x + 9 \).
We have
\[ \begin{align} x^2 - 16y^2 - 6x + 9 &= (x^2 - 6x + 9) - 16y^2 \\ &= (x-3)^2 - (4y)^2 \\ &= (x - 3 - 4y)(x - 3 + 4y). \ _\square \end{align} \]
\( x^3 + 3x^2 + 4x + 12 \) can be written as \( (x+a)(x^2+b) \) for some integers \(a\) and \(b.\)
What is \( b ? \)