Lesson 87: Factoring Polynomials by Grouping

New Concept

When a polynomial has four terms, make two groups and factor out the greatest common factor from each group.

What’s next

Next, you’ll apply this method to factor polynomials with four terms and then extend the technique to trinomials by rewriting them.

Property

When a polynomial has four terms, make two groups and factor out the greatest common factor from each group.

Explanation

Think of a four-term polynomial as a fun team-building exercise! First, split them into two pairs. Find what's common in the first pair and factor it out. Do the same for the second pair. If you did it right, both pairs will now share a common binomial expression, which you can factor out for the grand finale!

Examples

$3x^2 + 6xy + 5x + 10y = (3x^2 + 6xy) + (5x + 10y) = 3x(x + 2y) + 5(x + 2y) = (x + 2y)(3x + 5)$
$a^3 - 4a^2 + 3a - 12 = (a^3 - 4a^2) + (3a - 12) = a^2(a - 4) + 3(a - 4) = (a - 4)(a^2 + 3)$

Property

To factor a trinomial $ax^2 + bx + c$, first find two factors of the product $ac$ that have a sum equal to $b$. Then rewrite the trinomial as a four-term polynomial using these factors and factor by grouping.

Explanation

This is a magic trick to turn a tricky trinomial into an easy four-term polynomial we already know how to solve! Multiply 'a' and 'c', then find two secret numbers that multiply to that product and add up to 'b'. Use these numbers to split the middle term into two, then use your regular grouping skills to win!

Examples

To factor $x^2 + 3x - 10$, find factors of $1 \cdot (-10) = -10$ that sum to $3$. They are $5$ and $-2$.
$x^2 + 5x - 2x - 10 = x(x+5) - 2(x+5) = (x+5)(x-2)$
To factor $2k^2 - 7k + 6$, find factors of $2 \cdot 6 = 12$ that sum to $-7$. They are $-3$ and $-4$. So, $2k^2-3k-4k+6 = k(2k-3)-2(2k-3) = (2k-3)(k-2)$

Let's turn this three-term problem into a four-term puzzle we already know how to solve. We will factor a trinomial by grouping, a powerful application of our main concept.

Example Problem

Factor the trinomial $x^2 - 5x - 36$ by grouping.

Step-by-Step

1. First, identify the coefficients $a, b,$ and $c$ in $ax^2 + bx + c$. Here, $a=1, b=-5,$ and $c=-36$.
2. Find the product of $ac$.

3. Find two factors of $ac$ that have a sum equal to $b$. The factors of $-36$ that sum to $-5$ are $-9$ and $4$.

4. Rewrite the middle term, $-5x$, using these factors as $-9x + 4x$.

5. Now, group the four terms into two binomials and factor out the GCF from each.

6. Factor out the common binomial factor, $(x - 9)$.

Property

Remember to change the signs of terms within the parentheses when factoring out a negative one. For instance, $(b - a)$ can be rewritten as $-1(a - b)$.

Explanation

Ever get stuck when your binomials are perfect opposites, like $(y - 4)$ and $(4 - y)$? Don't worry! You can perform a 'sign flip' by factoring out a $-1$ from one of them. This turns the mismatched pair into an identical twin, letting you complete the grouping. It's a simple but powerful move that saves the day!

Examples

$5x(y - 2) + 4(2 - y) = 5x(y - 2) - 4(y - 2) = (y - 2)(5x - 4)$
$2a^2b - 10a + 25 - 5ab = (2a^2b - 10a) - (5ab - 25) = 2a(ab-5) - 5(ab-5) = (ab-5)(2a-5)$