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Factoring

Factoring a common scalar rewrites a sum as that scalar multiplied by the sum of the remaining factors.

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On this page 7 sections

Overview
Factoring collects a repeated multiplier outside parentheses
A common factor need not be the largest one
Check your understanding
Key takeaway
Sources & further reading
Concept connections

Figure 1Factoring 6x+9 as 3(2x+3). Three equal rows each contain a 2x block and a 3 block; widths are schematic because x is unspecified.

01Factoring collects a repeated multiplier outside parentheses#

Factoring collects a repeated multiplier outside parentheses. Start with $6x+9$ . Both terms contain a factor of $3$ .

Write that factor in each term.

6x+9=3\cdot 2x+3\cdot 3

The distributive rule works in reverse: the common $3$ multiplies the whole sum left inside.

Collect the common factor.

6x+9=3(2x+3)

Check by distributing. The outside factor must multiply every term, including the constant.

The original expression returns.

3(2x+3)=6x+9

Check your reasoning

Factor $4$ from $8x+12$ . Complete $4(2x+\square)$ .

Show answer and explanation

The constant must be $3$ : $4\cdot3=12$ . Putting $12$ inside would produce $48$ .

02A common factor need not be the largest one#

A common factor need not be the largest one. Both $2(6x+9)$ and $3(4x+6)$ equal $12x+18$ . A specified nonzero factor tells you what to divide out of each term.

A fixed scalar can multiply several terms.

5u+5v=5(u+v)

Check your reasoning

Pack $10a+15b$ into groups of $5$ . Which form works?

A $5(2a+3b)$
B $5(2a+15b)$
C $5(10a+3b)$

Show answer and explanation

$5(2a+3b)$

Divide both terms by $5$ : the group is $2a+3b$ .

03Check your understanding#

Check your reasoning

Repair $4x+8=4(x+8)$ .

A $4(x+2)$
B $4(4x+2)$
C $x+2$

Show answer and explanation

$4(x+2)$

The inside constant is $8/4=2$ . Keep the outside $4$ .

Key takeaway

Divide every term by the chosen factor inside parentheses. Distribute afterward to check that the value is unchanged.

Factor a common scalar factor from a sum

Sources & further reading

[1]
OpenStax Intermediate Algebra 2e, 6.1 ↗OpenStax · Book

01Factoring collects a repeated multiplier outside parentheses#

02A common factor need not be the largest one#

03Check your understanding#

Divide every term by the chosen factor inside parentheses. Distribute afterward to check that the value is unchanged.

Sources & further reading

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