The Distributive Property: Expanding and Factoring

Multiplying across a bracket

$$a(b + c) = ab + ac$$

The factor outside reaches every term inside.

$$3(x + 4) = 3x + 12$$ $$5(2n - 7) = 10n - 35$$

Multiplying only the first term is the error this property exists to prevent. $3(x + 4)$ is not $3x + 4$.

Why it works: the area model

A rectangle $3$ tall and $x + 4$ wide can be measured two ways.

	$x$	$4$
$3$	$3x$	$12$

As one rectangle, its area is $3(x + 4)$.

As two pieces, the areas are $3x$ and $12$, totalling $3x + 12$.

It is one rectangle either way, so the two must be equal:

$$3(x + 4) = 3x + 12$$

That is the whole justification. Nothing about it depends on the numbers chosen, which is why the property holds generally.

A negative outside the bracket

Every term inside changes sign.

$$-2(x - 5) = -2x + 10$$

Term by term:

$$-2 \times x = -2x \qquad -2 \times (-5) = +10$$

The second product is positive because two negatives multiply to a positive.

A lone minus in front of a bracket is a factor of $-1$:

$$-(x - 3) = -1(x - 3) = -x + 3$$

Subtracting a bracket flips every sign inside it. That is the single most common place this goes wrong, and writing the invisible $-1$ makes it visible.

Distributing comes before collecting

$$4(x + 2) + 3x$$

The bracket has to go first:

$$4x + 8 + 3x$$

Now the like terms can be collected:

$$7x + 8$$

Trying to combine $4(x+2)$ with $3x$ before expanding does not work, because the bracketed part is a single product rather than a term in $x$.

Factoring: the property backwards

$$ab + ac = a(b + c)$$

Find what every term has in common and pull it out.

$$6x + 9$$

Both terms divide by $3$:

$$3(2x + 3)$$

Check by distributing back: $3 \times 2x = 6x$ and $3 \times 3 = 9$. ✓

Expression	Common factor	Factored
$6x + 9$	$3$	$3(2x + 3)$
$10n - 15$	$5$	$5(2n - 3)$
$4x + 8y$	$4$	$4(x + 2y)$
$x^2 + 3x$	$x$	$x(x + 3)$

Take the greatest common factor. Factoring $6x + 9$ as $3(2x + 3)$ is finished; there is nothing left that both inside terms share.

Why having both directions matters

Expanding and factoring are the same rule run two ways, and each is useful for a different job.

Expanding clears brackets so terms can be collected and an equation solved. $3(x + 4) = 21$ becomes $3x + 12 = 21$, and from there the answer follows.

Factoring exposes structure that the expanded form hides. Writing $x^2 + 3x$ as $x(x + 3)$ shows immediately that the expression is zero when $x = 0$ or $x = -3$, which is the basis of solving quadratics by factoring.

The same rewrite is useful in plain arithmetic. Working out $7 \times 98$ mentally is awkward head-on and short work as $7(100 - 2) = 700 - 14 = 686$. The property is what licenses that split.

Worked examples

Common mistakes

Practice problems