4. Factoring Common Factors

Factoring Common Factors

Factoring is the reverse of expanding brackets. When we factor an expression, we look for a number, variable, or combination of both that is shared by all the terms.

For example:

6x + 12

Both terms are divisible by 6, so we can factor out 6:

6x + 12 = 6(x + 2)

Factoring allows us to rewrite expressions in a simpler or more useful form without changing their value.

Learning Targets

• I can identify common numerical factors.
• I can factor out common variables.
• I can write expressions in factored form.
• I can verify factoring by expanding.
• I can recognize when factoring simplifies a problem.

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What Is a Factor?

A factor is a number or algebraic quantity that is multiplied by another quantity.

For example:

3 × 4 = 12

Therefore, 3 and 4 are factors of 12.

The factors of 12 include:

1, 2, 3, 4, 6, 12

In algebra:

3x

can be thought of as:

3 × x

So both 3 and x are factors of 3x.

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Common Factors

A common factor is a factor shared by two or more terms.

Consider:

6x + 9

The numerical factors of 6 include:

1, 2, 3, 6

The numerical factors of 9 include:

1, 3, 9

The greatest factor shared by both is:

3

Therefore, 3 is the greatest common factor.

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Greatest Common Factor

The greatest common factor, often called the GCF, is the largest factor shared by all the terms.

Consider:

12 and 18

Factors of 12:

1, 2, 3, 4, 6, 12

Factors of 18:

1, 2, 3, 6, 9, 18

The greatest common factor is:

6

Therefore:

GCF = 6

Finding the GCF is useful because it allows us to factor an expression completely.

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Factoring Is the Reverse of Distribution

Recall the distributive property:

3(x + 4) = 3x + 12

Factoring simply reverses this process:

3x + 12 = 3(x + 4)

So:

Expansion → removes brackets

while:

Factoring → creates brackets

Understanding this connection makes factoring much easier.

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Factoring Out a Numerical Factor

Consider:

8x + 12

Both 8 and 12 are divisible by 4.

Factor out 4:

8x + 12 = 4(2x + 3)

Why?

Because:

8x ÷ 4 = 2x

and:

12 ÷ 4 = 3

Therefore:

4(2x + 3)

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A Simple Factoring Method

To factor an expression:

• Find a factor shared by every term.
• Place that factor outside brackets.
• Divide each original term by the common factor.
• Write the results inside the brackets.
• Expand your answer to check it.

For:

10x + 15

the greatest common factor is:

5

Divide each term by 5:

10x ÷ 5 = 2x

15 ÷ 5 = 3

Therefore:

10x + 15 = 5(2x + 3)

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Worked Example — Numerical Factor

Factor:

12x + 20

The GCF of 12 and 20 is:

4

Divide each term by 4:

12x ÷ 4 = 3x

20 ÷ 4 = 5

Therefore:

12x + 20 = 4(3x + 5)

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Worked Example — Subtraction

Factor:

15x − 25

The greatest common factor is:

5

Divide:

15x ÷ 5 = 3x

−25 ÷ 5 = −5

Therefore:

15x − 25 = 5(3x − 5)

Notice that the subtraction sign remains inside the brackets.

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Factoring Out Variables

Common factors can also be variables.

Consider:

x² + 4x

Both terms contain x.

Remember:

x² = x × x

and:

4x = 4 × x

So both terms contain a factor of x.

Factor out x:

x² + 4x = x(x + 4)

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Worked Example — Common Variable

Factor:

3x² + 6x

Both terms contain x.

We could write:

x(3x + 6)

This is correct, but we can factor further because 3 is also common.

Both terms share:

3x

Therefore:

3x² + 6x = 3x(x + 2)

Factoring out the greatest common factor usually gives the simplest factored form.

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Numerical and Variable Factors Together

Consider:

8x² + 12x

First examine the numbers.

The GCF of 8 and 12 is:

4

Now examine the variables.

Both terms contain:

x

Therefore, the greatest common factor is:

4x

Factor it out:

8x² ÷ 4x = 2x

12x ÷ 4x = 3

Therefore:

8x² + 12x = 4x(2x + 3)

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Finding the Common Variable Factor

When variables contain exponents, use the smallest power shared by every term.

Consider:

x³ + x²

Both terms contain:

x²

because:

x³ = x² × x

Therefore:

x³ + x² = x²(x + 1)

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Worked Example — Variables with Powers

Factor:

6x³ + 9x²

The numerical GCF is:

3

The smallest power of x shared by both terms is:

x²

Therefore, the GCF is:

3x²

Divide each term:

6x³ ÷ 3x² = 2x

9x² ÷ 3x² = 3

Therefore:

6x³ + 9x² = 3x²(2x + 3)

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More Than One Variable

Expressions may contain several variables.

Consider:

6xy + 9x

Both terms contain:

3x

Factor:

6xy ÷ 3x = 2y

9x ÷ 3x = 3

Therefore:

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

The variable y cannot be factored out because it does not appear in every term.

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Worked Example — Several Variables

Factor:

12x²y + 18xy²

The numerical GCF of 12 and 18 is:

6

Both terms contain x.

Both terms contain y.

Therefore, the GCF is:

6xy

Now divide:

12x²y ÷ 6xy = 2x

18xy² ÷ 6xy = 3y

Therefore:

12x²y + 18xy² = 6xy(2x + 3y)

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Factoring Three or More Terms

The same method works with several terms.

Consider:

6x + 12y + 18

The greatest numerical factor shared by all three terms is:

6

Therefore:

6x + 12y + 18 = 6(x + 2y + 3)

The common factor must be shared by every term.

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Worked Example — Several Terms

Factor completely:

8x² + 12x + 20x³

Look at the coefficients:

8, 12, 20

Their GCF is:

4

Every term also contains at least one x.

Therefore, the GCF is:

4x

Divide each term:

8x² ÷ 4x = 2x

12x ÷ 4x = 3

20x³ ÷ 4x = 5x²

Therefore:

8x² + 12x + 20x³ = 4x(2x + 3 + 5x²)

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Writing Expressions in Factored Form

An expression is in factored form when it is written as a product of factors.

For example:

6x + 18

is written in expanded form.

Its factored form is:

6(x + 3)

Similarly:

10x² + 15x

can be written:

5x(2x + 3)

Both forms represent the same value.

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Expanded Form and Factored Form

Consider:

4x + 20

Expanded form:

4x + 20

Factored form:

4(x + 5)

Neither form changes the value of the expression.

They are simply different ways of representing the same mathematical relationship.

Different forms can be useful for different types of problems.

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Checking Factoring by Expanding

One of the best ways to check factoring is to expand the answer.

Suppose we factor:

12x + 18

as:

6(2x + 3)

Check by distributing 6:

6 × 2x = 12x

6 × 3 = 18

Therefore:

6(2x + 3) = 12x + 18

We have returned to the original expression.

The factoring is correct.

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Worked Example — Checking

Suppose:

15x² + 10x = 5x(3x + 2)

Expand the right side:

5x × 3x = 15x²

5x × 2 = 10x

Therefore:

5x(3x + 2) = 15x² + 10x

The original expression has been recovered, so the factoring is correct.

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Factoring Out a Negative Number

Sometimes it is useful to factor out a negative value.

Consider:

−6x − 12

We could factor out:

−6

Then:

−6x ÷ −6 = x

and:

−12 ÷ −6 = 2

Therefore:

−6x − 12 = −6(x + 2)

Factoring out a negative value can make the expression inside the brackets easier to work with.

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Be Careful with Signs

Consider:

−8x + 12

Factor out −4:

−8x ÷ −4 = 2x

12 ÷ −4 = −3

Therefore:

−8x + 12 = −4(2x − 3)

Check by expanding:

−4(2x − 3)

= −8x + 12

Correct.

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Factoring and Area

Factoring can be understood geometrically.

Suppose the area of a rectangle is represented by:

6x + 12

We can factor:

6x + 12 = 6(x + 2)

This can represent a rectangle with dimensions:

6

and:

x + 2

Factoring can therefore help us determine possible dimensions from an area expression.

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Worked Example — Rectangle

Suppose the area of a rectangle is:

8x + 24

Find a possible pair of expressions for its dimensions.

Factor the expression:

8x + 24

The GCF is:

8

Therefore:

8x + 24 = 8(x + 3)

Possible dimensions are:

8

and:

x + 3

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Why Factor?

Factoring is useful because it can reveal the structure of an expression.

Consider:

12x + 24

In expanded form, we see two terms.

In factored form:

12(x + 2)

we can immediately see that the entire expression is a multiple of 12.

Factoring can also make:

• Calculations easier
• Equations easier to solve
• Fractions easier to simplify
• Patterns easier to recognize
• Geometric relationships easier to understand

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Factoring to Simplify Calculations

Consider:

17 × 6 + 17 × 4

We could calculate each multiplication separately:

102 + 68 = 170

But notice that 17 is a common factor.

Factor:

17(6 + 4)

Then:

17(10) = 170

Factoring has made the calculation easier.

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Another Mental Mathematics Example

Consider:

23 × 7 + 23 × 3

Factor out 23:

23(7 + 3)

= 23(10)

= 230

Recognizing the common factor reduces the number of calculations required.

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Factoring in Algebraic Fractions

Factoring can sometimes reveal factors that cancel in algebraic fractions.

For example:

(6x + 12) / 6

Factor the numerator:

6(x + 2) / 6

The common factor of 6 cancels:

x + 2

This is much easier to see after factoring.

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Factoring to Solve Equations

Factoring becomes especially useful when solving more advanced equations.

For example:

x² + 4x = 0

Factor out x:

x(x + 4) = 0

This form makes the structure of the equation much clearer.

Later, this idea can be used to solve quadratic equations.

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Choosing the Greatest Common Factor

You can factor out any common factor, but using the greatest common factor usually produces the simplest result immediately.

Consider:

12x + 18

We could factor out 2:

2(6x + 9)

This is correct.

But 6x + 9 still has a common factor of 3.

Factor again:

2 × 3(2x + 3)

which gives:

6(2x + 3)

It is more efficient to recognize that 6 is the GCF from the beginning.

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Worked Example — Factoring Completely

Factor completely:

24x²y + 36xy²

Find the numerical GCF:

GCF of 24 and 36 = 12

Find the common variables:

Both terms contain:

x

and:

y

Therefore, the GCF is:

12xy

Divide:

24x²y ÷ 12xy = 2x

36xy² ÷ 12xy = 3y

Therefore:

24x²y + 36xy² = 12xy(2x + 3y)

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A Reliable Factoring Strategy

When looking for a common factor, examine the expression in two parts.

First look at the numbers:

What is their greatest common factor?

Then look at the variables:

Which variables appear in every term?

If variables have exponents, use the smallest exponent shared by all terms.

Combine these to create the GCF.

Then divide every term by that GCF.

Finally, expand your answer to check.

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Recognizing When Factoring Helps

Factoring is particularly useful when you notice:

• Every term is divisible by the same number.
• Every term contains the same variable.
• Several terms share both numerical and variable factors.
• A calculation contains repeated multiplication.
• An algebraic fraction contains common factors.
• An equation has a common factor in every term.
• An area expression needs to be rewritten as possible dimensions.

Recognizing these patterns is an important algebra skill.

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Common Mistakes

Mistake: Choosing a factor that is not common to every term

Consider:

6x + 9

You cannot factor out 6 because 9 is not divisible by 6.

A correct factor is:

3

Therefore:

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

Mistake: Forgetting the variable

Consider:

8x² + 12x

Factoring only 4 gives:

4(2x² + 3x)

This is correct, but it is not fully factored.

Both terms also contain x.

The complete factorization is:

4x(2x + 3)

Mistake: Using the largest exponent

Consider:

x³ + x²

You cannot factor out x³ because the second term does not contain x³.

The greatest common variable factor is:

x²

Mistake: Dividing only one term

If you factor 5 from:

10x + 15

you must divide every term by 5.

Correct:

5(2x + 3)

Mistake: Sign errors when factoring negatives

For:

−6x + 12

factoring out −6 gives:

−6(x − 2)

not:

−6(x + 2)

Mistake: Not checking the answer

Expand your factored expression.

If it does not return to the original expression, something went wrong.

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Key Terms

Factor – a number or algebraic quantity multiplied by another quantity.

Common factor – a factor shared by two or more terms.

Greatest common factor (GCF) – the largest factor shared by all terms.

Factoring – rewriting an expression as a product of factors.

Factored form – an expression written as a product.

Expanded form – an expression written after multiplication has been distributed.

Coefficient – the numerical factor multiplying a variable.

Variable – a symbol representing a number or quantity.

Distributive property – the rule connecting expanded and factored forms.

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Check Your Understanding

1. Find the greatest common factor of:

12 and 20

2. Factor:

6x + 12

3. Factor:

15x + 25

4. Factor:

8x − 20

5. Factor completely:

x² + 7x

6. Factor completely:

6x² + 9x

7. Factor completely:

12x² + 18x

8. Factor completely:

10xy + 15x

9. Factor completely:

8x³ + 12x²

10. Factor completely:

18x²y + 24xy²

11. Factor:

−10x − 15

12. Write:

14x + 21

in factored form and then expand your answer to verify it.

13. Explain why:

4x² + 8x

is not completely factored as:

4(x² + 2x)

14. The area of a rectangle is:

10x + 30

Factor the expression to suggest possible dimensions for the rectangle.

15. Explain how factoring can make the following calculation easier:

19 × 8 + 19 × 2

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Key Takeaways

• A factor is a quantity multiplied by another quantity.
• A common factor appears in every term of an expression.
• The greatest common factor (GCF) is the largest factor shared by all terms.
• Common factors can include numbers, variables, or both.
• When variables contain powers, use the smallest exponent shared by every term.
• Factoring rewrites an expression as a product.
• Factoring is the reverse of the distributive property.
• Expanded and factored forms are equivalent representations of the same expression.
• Factoring out the GCF usually gives the simplest factored form.
• Factoring can be checked by expanding the answer.
• Factoring can simplify calculations, equations, algebraic fractions, and geometric relationships.
• Recognizing common factors is an important step toward more advanced algebra, including solving quadratic equations.

Did You Know?

Factoring is closely related to how computers handle complicated mathematical expressions. Computer algebra systems often search for common factors because factored expressions can require fewer calculations. In mathematics, science, engineering, and computer programming, recognizing a shared structure can make a complicated problem much easier to work with.