Taking Out the Common Factor

When several terms share the same factor, you can pull it outside a bracket and leave the rest inside. This one move collapses long expressions into short ones and makes fractions cancel that otherwise look impossible — and it applies wherever terms are joined by a plus or a minus sign.

Concept When to use it

Use it when terms are joined by addition.

Use it when terms are joined by subtraction.

Not needed for a plain multiplication — the factors are already together.

The idea is simple: take out what the terms have in common, and multiply it by what is left.

Example Two simple cases

Simplify $5x + 7x$ and $6xy + 7xy$ .

Both terms of the first contain

x

, so that is the common factor.

5x + 7x = x(5+7) = x(12) = 12x

In the second, both terms contain

xy

6xy + 7xy = xy(6+7) = xy(13) = 13xy

⟹ 12x and 13xy

Notice that the shared part never changes — only the numbers in front of it are combined.

Example Using it to simplify a fraction

Simplify $\dfrac{6xy + 7xy}{xy}$ .

Step 1 — factor the numerator:

6xy + 7xy = xy(6+7) = 13xy

Step 2 — rewrite the fraction:

\dfrac{13xy}{xy}

Step 3 — cancel the shared

xy

⟹ 13

The factor could only cancel after it was taken out. While the numerator was still a sum, nothing could be removed from it — a common factor has to multiply the whole numerator before it can cancel with the denominator.

Example Larger expressions, and subtraction

3x^2y + 5x^2y = x^2y(3+5) = 8x^2y

10ab - 3ab = ab(10-3) = 7ab

Subtraction behaves exactly like addition here — take the factor out and subtract inside the bracket instead. Take care to keep the terms in their original order, since $10 - 3$ and $3 - 10$ are not the same.

Note Finding the common factor

Look at what every term contains — letters as well as numbers.

Take out the largest part they all share.

Whatever remains of each term goes inside the bracket.

Note Mistakes to avoid

Cancelling part of a sum before factoring — nothing can be removed from

6xy + 7xy

until it becomes

13xy

Taking out only part of the shared factor, such as

x

when every term also has a

y

Forgetting to leave a 1 behind when a whole term is the factor, as in

x + 3x = x(1+3)

Reversing the order of a subtraction inside the bracket.

Adding the shared letters as well — in

6xy + 7xy

the

xy

stays as it is.

Summary

Taking out a common factor applies where terms are joined by addition or subtraction.
Pull out what every term shares and multiply it by what remains: 5x + 7x = x(5 + 7) = 12x.
The shared part is never added — only the coefficients combine.
Factoring is what allows a fraction to cancel, since a sum cannot be cancelled term by term.
Subtraction works the same way, but keep the terms in their original order inside the bracket.