Factor and Expand Linear Expressions

Students factor linear expressions by identifying the greatest common factor and rewriting the expression as a product (e.g., 6x − 9 = 3(2x − 3)). They also expand factored expressions using the distributive property with rational coefficients.

Try three of them

Real questions from this skill, graded here. No account, and nothing is saved.

Grade 7 · Expressions and Equations · question 1 of 3

0 of 20

20 rungs. A star at the top.

2(2x − 9) = 4x −

How it is taught

0:00 of 4:37

Expanding means distributing a factor across every term inside.

A negative sign in front distributes as -1.

Read it: A number times a difference

5(3x − 7) is a rectangle 5 tall: a part 3x wide, and a part 7 wide that is taken away. The 5 multiplies both parts: 5 × 3x = 15x and 5 × 7 = 35, so 5(3x − 7) = 15x − 35.

Draw the number outside the parentheses as the height, and what is inside as the width: the first part kept, the second taken away.
Multiply the height by each part: the part kept, then the part taken away.
To factor, find the height that goes into both parts, and divide each part by it.

The rectangle's area is the area of the part kept, take away the area of the part taken away.

The 5 is the height of both parts, so it multiplies the part taken away as well as the part kept.

Factoring runs it backwards: 15x − 35 = 5(3x − 7), because 15x ÷ 5 = 3x and 35 ÷ 5 = 7.

The number outside the parentheses multiplies every part inside, the part taken away too.

Multiplying only the first part: 5(3x − 7) written as 15x − 7.

The 5 multiplies the part taken away too: 5 × 7 = 35, so 5(3x − 7) = 15x − 35.

9(3x − 2) = 27x − ___ — 18. The 9 multiplies the part taken away too: 9 × 2 = 18.

Factor 6x − 9

Today we'll factor a linear expression. Factoring is like running multiplication backwards — we find a common factor and pull it out.

Here's our expression: six x minus nine.

What's the biggest number that divides both six and nine evenly?

Three! Three is the greatest common factor of six and nine.

We rewrite each term using three. Six x is three times two x, and nine is three times three.

Now we pull the three out front — that's the distributive property in reverse.

Grade 7 · Pull out the greatest common factor

That gives us three times the quantity two x minus three. Our expression is factored!

Let's check by expanding it back. We distribute the three to each term inside the parentheses.

Three times two x is six x.

Three times three is nine. We're back to six x minus nine — it works!

So six x minus nine factors as three times the quantity two x minus three. Nice work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

More of the library

Every skill has a page

Practise this skill free

14 days of every skill, practice packets, Fluency sheets and up to four students. No card.

Start free