Grade by Grade
C. Algebra · Algebra: Expressions, Equations and Coding

The Distributive Property: Multiplying Monomials, Expanding and Common Factors

C1.4 C1.3 A1About 70 minutes
Learning goals
  • I can multiply monomials, such as (3x)(4x) = 12x².
  • I can expand a product of a monomial and a polynomial using the distributive property and an area model.
  • I can expand and simplify expressions such as 3(x + 2) − 2(x − 5).
  • I can factor out a common factor and check by expanding.

Words to know

distributive property
a(b + c) = ab + ac. The factor outside the brackets multiplies every term inside.
expand
Remove brackets by multiplying, for example 3(x + 4) = 3x + 12.
area model
A rectangle whose side lengths are the factors and whose area is the product. It shows each part of a multiplication.
factor
(verb) Write an expression as a product. 6x + 9 = 3(2x + 3) is in factored form.
greatest common factor (GCF)
The largest monomial that divides every term. The GCF of 4x² and 8x is 4x.

Remember from Grade 8

In Grade 8 you used the distributive property with numbers: 7 × 4.98 = 7 × 5 − 7 × 0.02. Now the same property will let you multiply expressions with variables.

Let's start in the garden

A community garden plot is x metres wide. Its length is 3 m more than twice the width, so the length is 2x + 3 metres.

Area = width × length = x(2x + 3).

To work with this area, we want it without brackets. To do that, we need two skills: multiplying monomials, and the distributive property.

Multiplying monomials

To multiply monomials, multiply the coefficients and multiply the variable parts.

  • (3x)(4x) = (3 × 4)(x × x) = 12x²
  • (2x)(−5) = −10x
  • (−2y)(5y²) = −10 × y × y² = −10y³
  • (4a)(−3b) = −12ab

When you multiply powers with the same base, add the exponents: x × x = x¹ × x¹ = x², and y × y² = y³. (You work with these exponent rules for numbers in the Number strand too.)

Compare with adding: x + x = 2x, but x × x = x².

The distributive property with variables

The factor outside the brackets multiplies every term inside. An area model shows why.

x433 × x = 3x3 × 4 = 123(x + 4) = 3x + 12
A rectangle 3 units tall and (x + 4) units long splits into two parts: 3x and 12. Total area: 3(x + 4) = 3x + 12 (not to scale).
x+ 5xx2x2 x²-tiles10 x-tiles2x(x + 5) = 2x² + 10x
A rectangle 2x tall and (x + 5) long, built from algebra tiles. It uses 2 x²-tiles and 10 x-tiles, so 2x(x + 5) = 2x² + 10x.
Worked example

Example 1: Expand with a negative

Expand −4(2x − 3).

  1. Multiply −4 by each term inside: (−4)(2x) and (−4)(−3).
  2. (−4)(2x) = −8x.
  3. (−4)(−3) = +12 (a negative times a negative is positive).
  4. Check with x = 1: −4(2 − 3) = −4(−1) = 4, and −8 + 12 = 4 ✓.

−8x + 12

Worked example

Example 2: The garden area

Expand the garden area x(2x + 3). Then find the area when x = 4 m and check.

  1. x × 2x = 2x². x × 3 = 3x.
  2. Area = 2x² + 3x.
  3. When x = 4: 2(16) + 3(4) = 32 + 12 = 44 m².
  4. Check: width 4 m, length 2(4) + 3 = 11 m; 4 × 11 = 44 m² ✓.

2x² + 3x square metres; 44 m² when x = 4.

Worked example

Example 3: Expand and simplify

Simplify 3(x + 2) − 2(x − 5).

  1. Expand each part. 3(x + 2) = 3x + 6.
  2. −2(x − 5) = −2x + 10. (The −2 multiplies both terms, and (−2)(−5) = +10.)
  3. Collect like terms: 3x − 2x + 6 + 10 = x + 16.
  4. Check with x = 0: 3(2) − 2(−5) = 6 + 10 = 16, and 0 + 16 = 16 ✓.

x + 16

Worked example

Example 4: A trinomial

Expand x(x² − 2x + 3).

  1. x × x² = x³. x × (−2x) = −2x². x × 3 = 3x.

x³ − 2x² + 3x

Factoring: the distributive property backwards

Sometimes you want to go the other way: from a sum to a product. This is called factoring.

  1. Find the greatest common factor (GCF) of all the terms, including variables.
  2. Write the GCF outside the brackets.
  3. Divide each term by the GCF to find what goes inside.
  4. Check by expanding.

6x + 9: the GCF of 6 and 9 is 3, so 6x + 9 = 3(2x + 3). Check: 3(2x + 3) = 6x + 9 ✓.

4x² − 8x: the GCF of 4x² and 8x is 4x, so 4x² − 8x = 4x(x − 2). Check: 4x(x) − 4x(2) = 4x² − 8x ✓.

Watch out!

  • Multiply the outside factor by every term, including the last one.
  • Watch the signs: −2(x − 5) = −2x + 10.
  • After factoring, expand your answer to check it. If you get back the original expression, you factored correctly.
Hands-on activity

Build rectangles to multiply

You need: your algebra tiles (from the lesson on writing expressions), or paper squares and strips cut to size.

  1. Build a rectangle that is 2 units tall and (x + 3) long: lay 1 x-tile on its side with 3 unit tiles after it, then make an identical row underneath. What expression does the whole rectangle show? (2x + 6)
  2. Build a rectangle that is x tall and (x + 4) long: 1 x²-tile with 4 x-tiles standing up beside it. What is x(x + 4)? (x² + 4x)
  3. Build a rectangle that is 2x tall and (x + 2) long: make an x-by-(x + 2) rectangle (1 x²-tile and 2 x-tiles), then stack an identical one under it. Count the tiles. (2x² + 4x)
  4. Now go backwards: take 3 x²-tiles and 6 x-tiles. Can you arrange them into a rectangle? What are its side lengths? (Stack 3 rows of 1 x²-tile and 2 x-tiles: the rectangle is 3x by (x + 2), so 3x² + 6x = 3x(x + 2). A single long row, x by (3x + 6), also works.)

What to look for: the side lengths of each rectangle are the factors, and the total tiles are the expanded form.

Just for fun

The distributive property is like handing out snacks to a group: the number outside the brackets has to give one to every term inside. Skip one, and somebody gets grumpy (and your answer is wrong).

Your turn

1

Multiply (5x)(3x).

2

Multiply (−2a)(7a²).

3

Expand 4(2x − 3).

4

Expand −3(x + 6).

5

Expand x(x − 7).

6

Expand and simplify 2(x + 4) + 3(x − 1).

7

Factor 5x + 20.

More practice: warm-up

1

Expand 2(x + 5).

2

Multiply (4)(3x).

3

Multiply x · x.

4

Expand 3(a − 2).

5

Expand −(2x + 1).

More practice: on your own

1

Expand 3x(2x − 5).

2

Expand −2x(x² − 3x + 4).

3

Expand and simplify 5(2x − 1) − 3(x − 4).

4

Factor 8x² − 12x completely.

5

A rectangle is x cm wide and x + 6 cm long. Write its area in expanded form and evaluate for x = 5.

Challenge

1

Expand and simplify 2x(x − 3) − x(x + 4). Check with x = 1.

2

Show that 4(x + 3) − 2(2x − 1) has the same value for every x. What is that value?

3

Factor 6x²y − 9xy² completely and check.

Check your understanding

Quiz

1.

Multiply (3x)(−4x).

2.

Expand 5(x − 2).

3.

Expand −2(3x − 4).

4.

Expand x(x + 9).

5.

Simplify 2(x + 3) + 4(x − 1).

6.

Which is a correct factored form of 12x + 18?

For parents and teachers

Big idea: C1.4 asks students to simplify expressions by applying properties of operations. The distributive property is the most important one: it lets students expand products and factor sums. Area models and algebra tiles show why it works.

Common difficulties:

  • Multiplying only the first term: 4(2x − 3) written as 8x − 3.
  • Sign errors with a negative outside: −2(x − 5) written as −2x − 10.
  • Mixing up x + x = 2x and x × x = x².
  • Factoring out a factor that is not the greatest. 12x + 18 = 2(6x + 9) is correct but not complete; 6(2x + 3) is fully factored.

How to help: Ask your teen to check every expansion by substituting a small number (like x = 2) into both forms. Building rectangles with tiles is especially helpful for visual learners.