Factoring by Grouping

We can sometimes factor a difficult-looking polynomial through the creative use of the distributive property. This is called factoring by grouping.

Factoring by grouping in action

The polynomial $2 x y - 6 x z + 3 y - 9 z$ looks a little intimidating, but the coefficients have a proportional relationship we can exploit. The 2 and -6 are both divisible by 2, and both terms have an $x$ . That means we can use the distributive property to factor out $2x$ from both sides of the first two terms:

$2 x y - 6 x z = 2 x (y - 3 z)$

Similarly, we can factor out a 3 from the other two terms:

$3 y - 9 z = 3 (y - 3 z)$

The quantity $y - 3 z$ appears in both factored terms, allowing us to use the distributive property one more time to get a simplified answer of:

$2 x (y - 3 z) + 3 (y - 3 z) = (2 x + 3) (y - 3 z)$

Factoring by grouping practice question

Factor $x^{2} + x y + 3 x + 3 y$

We can begin by grouping the terms as follows:

$(x^{2} + 3 x) (x y + 3 y)$

Both terms have $x + 3$ as a factor, so we can use the distributive property to simplify

$x (x + 3) + y (x + 3)$

into a final answer of

$(x + y) (x + 3)$

Topics related to the Factoring by Grouping

Advanced Factoring

Solving Quadratic Equations using Factoring

Graphing Quadratic Equations Using Factoring

Flashcards covering the Factoring by Grouping

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Factoring by Grouping

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

Get help with factoring by grouping

Factoring by grouping is seldom the only way to factor a polynomial, but it's generally the easiest, fastest method when available. If your student isn't taking full advantage of it, they may not have enough time to complete exams or spend longer than needed on homework. Luckily, an experienced mathematics tutor could provide practice problems until factoring by grouping feels like second nature. Reach out to the Educational Directors at Varsity Tutors today to learn more.