How to use the distributive property with parentheses

If a number multiplies a whole expression in parentheses, we can multiply it by each term separately. Let’s start with two terms, then see how the signs work when the outside factor is negative.

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Multiply the outside factor by each term

$\begin{array}{l}a(b+c)=ab+ac\\ a(b-c)=ab-ac\end{array}$

Subtracting $c$ is the same as adding $-c$, so the two rules use the same idea. If we look at $4(3-x)$ as $4(3+(-x\left)\right)$, the terms inside are $3$ and $-x$.

Multiply each term by 4, then add the two products.

$\begin{array}{l}4(3-x)=4\left(3\right)+4(-x)\\ =12-4x\end{array}$

We can check the result with an input. At $x=2$, the original expression is $4(3-2)=4$, and the distributed expression is $12-4\left(2\right)=4$.

Example with a negative fraction outside the parentheses

Multiply $-\frac{1}{3}$ by $3y$ and by $15$. The $3$ in the first term cancels with the denominator, and $15$ divided by $3$ is $5$. Both products are negative multiples of the original terms.

$\begin{array}{l}-\frac{1}{3}(3y+15)\\ =-\frac{1}{3}\left(3y\right)-\frac{1}{3}\left(15\right)\\ =-y-5\end{array}$

Example with three terms

With three terms, we make three products. In $-2(x^2-3x+4)$, the middle term is $-3x$, so multiplying it by $-2$ gives $6x$.

$\begin{array}{l}-2(x^2-3x+4)\\ =-2x^2+6x-8\end{array}$

The terms $x^2$, $x$, and the constant are different kinds of terms, so this expression is already simplified.

When both factors have parentheses, see how to multiply two binomials with the distributive property.

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