Factor quadratics using a common factor

Ontario Grade 10 Mathematics

Quadratic Relations

Find what every term shares, then write the expression as a product

A quadratic expression has a highest variable exponent of

2

. For example,

6x^2+9x

is quadratic. Factoring means rewriting an expression as a product. In this lesson, you will factor a quadratic by finding a factor shared by every term. This reverses the distributive property. Look at the terms, find what they share, and check your result by expanding.

What you will learn

Identify the terms in a quadratic expression and find their greatest common factor.
Rewrite a quadratic expression as a product by factoring out its common factor.
Check a factorization by expanding the product.

1. Review: terms, factors, and the distributive property

An expression is a mathematical phrase without an equals sign. In

6x^2+9x

, the terms are

6x^2

and

9x

. A plus or minus sign separates terms. Keep the sign with the term that follows it when working with subtraction.

A factor is a number or variable multiplied by another number or variable. In

6x^2

, the factors include

6

,

x

, and

x

, because

x^2

means

x\times x

. The coefficient is the number multiplying the variable; the coefficient of

6x^2

is

6

.

The distributive property lets you multiply a factor by each term inside brackets. For example,

3(x+4)=3x+12

. Factoring out a common factor reverses this action. The common factor must be a factor of every term.

a(b+c)=ab+ac

A quadratic expression has a highest variable exponent of $2$ .
Factoring rewrites an expression as a product.
Expanding brackets can check whether a factorization is correct.

2. Find the greatest common factor

The greatest common factor, or GCF, is the largest positive factor shared by all the terms. To find it, first look for a number that divides every coefficient. Then look for any variable that appears in every term. When a variable appears in each term, use the smallest exponent it has in those terms.

For example, the terms

12x^2

and

18x

share the number factor

6

. They also both contain at least one

x

. Their GCF is therefore

6x

. The second term has only one

x

, so the common variable factor is

x

, not

x^2

.

If the terms have no variable in common, the GCF may be a number only. For this lesson, focus on expressions where the terms share a factor.

\operatorname{GCF}(12x^2,18x)=6x

The GCF is stated as a positive factor that divides every term exactly.
Use the smallest exponent of a shared variable.
You can choose the negative of the GCF when factoring if that makes the first term inside the brackets positive.

3. Factor out the common factor

Divide each term by the GCF to find what belongs inside the brackets. Put the GCF outside the brackets. The terms inside are the quotients: each original term divided by the common factor.

For

12x^2+18x

, dividing each term by

6x

gives

2x

and

3

. So the expression becomes

6x(2x+3)

. This works because multiplying

6x

by each term in the brackets returns the original terms.

With subtraction, keep track of the negative sign. For

-8x^2-12x

, the positive GCF is

4x

. You may choose to factor out

-4x

instead; this leaves

2x+3

inside the brackets. Choosing the negative factor makes the first term inside positive.

ab+ac=a(b+c)

Divide every term by the same GCF, or by its negative if that is your chosen factor.
Place the quotients inside one set of brackets.
Check signs by multiplying back.

4. Check a factorization by expanding

After factoring, use the distributive property to check your work. Multiply the outside factor by each term inside the brackets. The result should match the original expression, including its signs and exponents.

For instance, expanding

5x(2x-7)

gives

10x^2-35x

. That matches the original expression, so the factorization is correct. If the result has a different coefficient or sign, revisit the division of the terms by the common factor.

This check is especially useful when the terms contain negative signs. It also confirms that no term was left out.

a(b-c)=ab-ac

A correct factorization expands to the original expression.
Check every term, coefficient, variable exponent, and sign.

Organize the quotients when factoring

Original term	Divide by the common factor $5x$	Quotient
$15x^2$	$15x^2\div 5x$	$3x$
$25x$	$25x\div 5x$	$5$

Worked example

Example 1: Factor a quadratic with a variable GCF

Factor

15x^2+25x

using a common factor.

Find the shared factor
The coefficients $15$ and $25$ share the factor $5$ . Both terms also contain $x$ . The GCF is $5x$ .
$\operatorname{GCF}(15x^2,25x)=5x$
Divide each term
Divide $15x^2$ by $5x$ to get $3x$ . Divide $25x$ by $5x$ to get $5$ . These quotients go inside the brackets.
$15x^2\div 5x=3x,\qquad 25x\div 5x=5$
Write the product
Put the GCF outside the brackets and the quotients inside. This gives a product equal to the original quadratic expression.
$15x^2+25x=5x(3x+5)$

Answer:

5x(3x+5)

Check: Expand to check:

5x(3x+5)=15x^2+25x

, which matches the original expression.

Worked example

Example 2: Factor when both terms are negative

Factor

-14x^2-21x

using a common factor.

Find the GCF and choose a factor
The positive GCF is $7x$ : the coefficients share $7$ , and both terms contain $x$ . Since both terms are negative, choose the negative factor $-7x$ for the factorization so the first term inside the brackets is positive.
$\operatorname{GCF}(-14x^2,-21x)=7x$
Divide each term
Divide $-14x^2$ by $-7x$ to get $2x$ . Divide $-21x$ by $-7x$ to get $3$ . Both quotients are positive.
$(-14x^2)\div(-7x)=2x,\qquad (-21x)\div(-7x)=3$
Write the product
Place the chosen factor $-7x$ outside the brackets and the quotients inside. Multiplying back restores both negative terms.
$-14x^2-21x=-7x(2x+3)$

Answer:

-7x(2x+3)

Check: Expand to check:

-7x(2x+3)=-14x^2-21x

, which matches the original expression.

Common mistakes and how to avoid them

Calling a negative factor the GCF.

Correction: State the GCF as positive. You may choose its negative as the factor you take out when forming the product.

Taking out a factor that does not divide every term.

Correction: Test the proposed factor against each term. A common factor must divide every term exactly.

Using

x^2

as the common variable factor in

12x^2+18x

.

Correction: The second term contains only one

x

. Use the smallest shared exponent, so the variable part of the GCF is

x

.

Forgetting a negative sign when dividing terms.

Correction: Divide each signed term by the chosen signed factor, then expand your answer to check the signs.

Leaving a term out of the brackets.

Correction: Divide every term in the original expression by the chosen factor. Each quotient must appear inside the brackets.

Lesson summary

Identify every term in the quadratic expression.
Find the positive greatest common factor shared by all terms.
Divide every term by the factor you choose to take out, then place the quotients in brackets.
Expand the result to confirm that it matches the original expression.

Check your understanding

Question 1

What is the GCF of

18x^2

and

30x

?

$6x$
$6x^2$
$3x$
$6$

Show answer and explanation

6x

The coefficients share a factor of

6

, and both terms contain one

x

. The smallest shared variable factor is

x

, so the GCF is

6x

.

Question 2

Which expression is the factorization of

8x^2+20x

using a common factor?

$4x(2x+5)$
$4x(2x+20)$
$2x(4x+20)$
$4(2x^2+5x)$

Show answer and explanation

4x(2x+5)

The GCF is

4x

. Dividing the terms by

4x

gives

2x

and

5

, so the factorization is

4x(2x+5)

. Expanding it returns

8x^2+20x

.

Question 3

Factor

-9x^2-12x

by taking out a negative common factor.

$-3x(3x+4)$
$-3x(3x-4)$
$-3(3x^2+4x)$
$-9x(x+12)$

Show answer and explanation

-3x(3x+4)

Taking out

-3x

leaves

3x

and

4

. Expanding

-3x(3x+4)

gives

-9x^2-12x

.

Key terms

Term: A part of an expression separated from other parts by a plus or minus sign.
Factor: A number or variable multiplied by another number or variable.
Coefficient: The number multiplying the variable in a term.
Greatest common factor (GCF): The largest positive factor shared by every term in an expression.
Factoring: Rewriting an expression as a product of factors.

Continue through MFM2P

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic Q3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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