4.4Expansion: The Distributive Law

Why does $3 \times 12 = 3 \times 10 + 3 \times 2$ ? Because multiplication distributes over addition. This is the distributive law, and it is the engine that powers expansion in algebra.

Distributive law

a(b + c) = ab + ac

To expand an expression, multiply the term outside the bracket by every term inside.

- 4 (3 y + 7)

- 4 \times 3 y

= - 12 y

- 4 \times 7

= - 28

both arrows start at the same −4

The

-4

multiplies both terms inside the bracket, sign and all.

Single bracket expansion

Expand $\displaystyle -4(3y + 7)$ .

Show solution▾

\displaystyle -4(3y + 7)

\displaystyle = (-4) \times 3y + (-4) \times 7

\displaystyle = -12y - 28

Now you try

Expand $6(x - 9)$ .

Show solution▾

\displaystyle 6(x - 9)

\displaystyle = 6 \times x - 6 \times 9 \quad \scriptsize\textit{distributive law}

\displaystyle = 6x - 54

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Two brackets

Expand and simplify $\displaystyle 5(2a - b) - 3(4a - 7b)$ .

Show solution▾

= 10a - 5b - 12a + 21b \quad \scriptsize\textit{expand each bracket}

= (10 - 12)a + (-5 + 21)b \quad \scriptsize\textit{group like terms}

= -2a + 16b

Now you try

Expand and simplify $5(2x + y) - 2(2x + 2y)$ .

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= 10x + 5y - 4x - 4y \quad \scriptsize\textit{expand}

= (10 - 4)x + (5 - 4)y

= 6x+y

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Predict

After expanding and simplifying $3(3x - 1) + k(x + 5)$ , the result is $11x + 7$ . What is $k$ ?

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Expand:

3(3x - 1) + k(x + 5) = 9x - 3 + kx + 5k

= (9 + k)x + (5k - 3)

You do not need to solve for

x

. Compare the coefficients of

x

9 + k = 11

k = 2

Check with the constant terms:

5(2) - 3 = 7

✓