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6. Polynomials & Factoring
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United States of AmericaFL
Grade 912

6.06 Factoring by grouping

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Lesson

Concept summary

Polynomials with four terms can sometimes be factored using a method called factoring by grouping. This method involves grouping the terms into two pairs and taking a common factor out of each pair, then taking a common binomial factor out of the two resulting terms.

\displaystyle ax + ay + bx + by\displaystyle =\displaystyle a\left(x + y\right) + b\left(x + y\right)
\displaystyle =\displaystyle \left(x+y\right)\left(a+b\right)

Sometimes the terms will need to be rearranged before common factors can be taken out of each pair.

It is also possible to factor some polynomials with more than four terms using this method - for instance, a polynomial with six terms might be able to split into three pairs which each leave behind a common binomial.

Note that not every four-term polynomial is factorable in this way. For instance, the polynomial3x + 6y + x + 5zdoesn't have enough common factors between any possible pairing of terms.

Worked examples

Example 1

Factor the expression 10 xy + 4x + 15y + 6.

Approach

We arrange the terms first, grouping those with common factors. We factor out the GCF on each pair and the common binomial factor afterwards.

Solution

\displaystyle 10 xy + 4x + 15y + 6\displaystyle =\displaystyle \left(10 xy + 4x\right)+ \left(15y + 6\right)Split based on common factors
\displaystyle =\displaystyle 2x\left(5y + 2\right)+ 3\left(5y + 2\right)Factor out each GCF \left(2x \text{ and } 3 \right)
\displaystyle =\displaystyle \left(5y + 2\right)\left(2x + 3\right)Factor out the common binomial factor

Since \left(5y + 2\right)\left(2x + 3\right) cannot be factored further, it is the final answer.

Reflection

  • Alternatively, we can group 10xy \text{ and } 15y and 4x \text{ and } 6 together and get the same answer.

    \displaystyle 10 xy + 4x + 15y + 6\displaystyle =\displaystyle \left(10 xy + 15y\right)+ \left(4x + 6\right)Split based on common factors
    \displaystyle =\displaystyle 5y\left(2x + 3\right)+ 2\left(2x + 3\right)Factor out each GCF \left(5y \text{ and } 2 \right)
    \displaystyle =\displaystyle \left(2x + 3\right)\left(5y + 2\right)Factor out the common binomial factor

  • We can check the answer by multiplying the factored form \left(5y + 2\right)\left(2x + 3\right).

    \displaystyle \left(5y + 2\right)\left(2x + 3\right)\displaystyle =\displaystyle 2x\left(5y + 2\right) + 3\left(5y + 2\right)Distribute \left(5y + 2\right)
    \displaystyle =\displaystyle 10xy + 4x + 15y + 6Check

Example 2

Factor the expression 4a^{2} + 5ab - 8a - 10b.

Approach

We arrange the terms first, grouping those with common factor. We factor out the GCF on each pair and the common binomial factor afterwards.

Solution

\displaystyle 4a^{2} + 5ab - 8a - 10b\displaystyle =\displaystyle \left(4a^{2} + 5ab \right)+ \left(- 8a - 10b\right)Split based on common factors
\displaystyle =\displaystyle a\left(4a+5b\right) - 2\left(4a+5b\right)Factor out the GCF \left(a \text{ and } -2 \right)
\displaystyle =\displaystyle \left(4a + 5b\right)\left(a - 2\right)Factor out the common binomial factor

Since \left(4a + 5b\right)\left(a - 2\right) cannot be factored further, it is the final answer.

Reflection

  • Alternatively, we can group 4a^{2} \text{ and } - 8a and 5ab \text{ and } - 10b together and get the same answer.

    \displaystyle 4a^{2} + 5ab - 8a - 10b\displaystyle =\displaystyle \left(4a^{2} - 8a \right)+ \left(5ab - 10b\right)Split based on common factors
    \displaystyle =\displaystyle 4a\left(a- 2\right) + 5b\left(a-2\right)Factor out the GCF \left(4a \text{ and } 5b \right)
    \displaystyle =\displaystyle \left(a - 2\right)\left(4a + 5b\right)Factor out the common binomial factor

  • We can check the answer by multiplying the factored form \left(4a + 5b\right)\left(a - 2\right).

    \displaystyle \left(4a + 5b\right)\left(a - 2\right)\displaystyle =\displaystyle a\left(4a + 5b\right) -2\left(4a + 5b\right)Distribute \left(4a + 5b\right)
    \displaystyle =\displaystyle 4a^{2} + 5ab - 8a- 10bCheck

Example 3

Factor the expression 3x^{3} + 3x^{2} - 6x - 6.

Approach

We arrange the terms first, grouping those with common factor. We factor out the GCF on each pair and the common binomial factor afterwards.

Solution

\displaystyle 3x^{3} + 3x^{2} - 6x - 6\displaystyle =\displaystyle \left(3x^{3} + 3x^{2} \right)+ \left(- 6x - 6\right)Split based on common factors
\displaystyle =\displaystyle 3x^2\left(x+1\right) - 6\left(x+1\right)Factor out the GCF \left(3x^{2} \text{ and } -6 \right)
\displaystyle =\displaystyle \left(x+1\right)\left(3x^{2} - 6\right)Factor out the common binomial factor
\displaystyle =\displaystyle 3\left(x+1\right)\left(x^{2} - 2\right)Factor out 3 in 3x^{2} - 6

Since 3\left(x+1\right)\left(x^{2} - 2\right) cannot be factored further, it is the final answer.

Reflection

  • Alternatively, we can also factor out 3 at the start.

  • We can check the answer by multiplying the factored form 3\left(x+1\right)\left(x^{2} - 2\right).

    \displaystyle 3\left(x+1\right)\left(x^{2} - 2\right)\displaystyle =\displaystyle 3\left(x^{2}\left(x+1\right) - 2\left(x+1\right)\right) Distribute \left(x + 1\right)
    \displaystyle =\displaystyle 3\left(x^{3}+x^{2} - 2x-2\right)Multiply
    \displaystyle =\displaystyle 3x^{3} + 3x^{2} - 6x - 6Check

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.1.7

Rewrite a polynomial expression as a product of polynomials over the real number system.

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