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How to Expand Parentheses with Variables

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When you’re working with both variables and parentheses, you need to follow a particular set of rules. These rules allow you to remove the parentheses to clean up your expression. In school, you’ll spend a good amount of time working with expressions like this, so make sure to pay close attention.

Rule

Numbers and Signs Multiplied by Parentheses

1.: A $+$ in front of a parenthesis changes nothing inside the parentheses.
2.: A $-$ in front of a parenthesis changes all the signs inside the parentheses, so a $+$ becomes a $-$ and a $-$ becomes a $+$ .
3.: A positive factor multiplied by a parenthesis leaves the signs unchanged, but the terms inside the parentheses are multiplied by the factor outside.
4.: A negative factor multiplied by a parenthesis changes all the signs inside the parentheses, and the terms inside the parentheses are multiplied by the factor outside.

Here are a couple of examples illustrating the four rules above:

Example 1

Rule 1

Expand the parentheses:

+ (x + 1) = x + 1

Expand the parentheses:

+ (x - 1) = x - 1

Expand the parentheses:

+ (- x - 1) = - x - 1

Expand the parentheses:

+ (- x + 1) = - x + 1

Example 2

Rule 2

Expand the parentheses:

\begin{array}{l} - (x + 1) & = - + x - + 1 \\ = - x - 1 \end{array}

\begin{array}{l} - (x + 1) & = - + x - + 1 \\ = - x - 1 \end{array}

Expand the parentheses:

\begin{array}{l} - (x - 1) & = - + x - - 1 \\ = - x + 1 \end{array}

\begin{array}{l} - (x - 1) & = - + x - - 1 \\ = - x + 1 \end{array}

Expand the parentheses:

\begin{array}{l} - (- x - 1) & = - - x - - 1 \\ = x + 1 \end{array}

\begin{array}{l} - (- x - 1) & = - - x - - 1 \\ = x + 1 \end{array}

Expand the parentheses:

\begin{array}{l} - (- x + 1) & = - - x - + 1 \\ = x - 1 \end{array}

\begin{array}{l} - (- x + 1) & = - - x - + 1 \\ = x - 1 \end{array}

Example 3

Rule 3

\begin{array}{l} 2 a (a + 3) & = + 2 a \cdot (+ a) + 2 a \cdot (+ 3) \\ = 2 a^{2} + 6 a \end{array}

\begin{array}{l} 2 a (a + 3) & = + 2 a \cdot (+ a) + 2 a \cdot (+ 3) \\ = 2 a^{2} + 6 a \end{array}

Expand the parentheses:

\begin{array}{l} 2 a (a - 3) & = + 2 a \cdot (+ a) + 2 a \cdot (- 3) \\ = 2 a^{2} - 6 a \end{array}

\begin{array}{l} 2 a (a - 3) & = + 2 a \cdot (+ a) + 2 a \cdot (- 3) \\ = 2 a^{2} - 6 a \end{array}

Expand the parentheses:

\begin{array}{l} 2 a (- a - 3) & = + 2 a \cdot (- a) + 2 a \cdot (- 3) \\ = - 2 a^{2} - 6 a \end{array}

\begin{array}{l} 2 a (- a - 3) & = + 2 a \cdot (- a) + 2 a \cdot (- 3) \\ = - 2 a^{2} - 6 a \end{array}

Expand the parentheses:

\begin{array}{l} 2 a (- a + 3) & = + 2 a \cdot (- a) + 2 a \cdot (+ 3) \\ = - 2 a^{2} + 6 a \end{array}

\begin{array}{l} 2 a (- a + 3) & = + 2 a \cdot (- a) + 2 a \cdot (+ 3) \\ = - 2 a^{2} + 6 a \end{array}

Example 4

Rule 4

Expand the parentheses:

\begin{array}{l} - a (b + a) & = a \cdot (+ b) - a \cdot (+ a) \\ = - a b - a^{2} \end{array}

\begin{array}{l} - a (b + a) & = - a \cdot (+ b) - a \cdot (+ a) \\ = - a b - a^{2} \end{array}

Expand the parentheses:

\begin{array}{l} - a (b - a) & = - a \cdot (+ b) - a \cdot (- a) \\ = - a b + a^{2} \end{array}

\begin{array}{l} - a (b - a) & = - a \cdot (+ b) - a \cdot (- a) \\ = - a b + a^{2} \end{array}

Expand the parentheses:

\begin{array}{l} - a (- b - a) & = - a \cdot (- b) - a \cdot (- a) \\ = a b + a^{2} \end{array}

\begin{array}{l} - a (- b - a) & = - a \cdot (- b) - a \cdot (- a) \\ = a b + a^{2} \end{array}

Expand the parentheses:

\begin{array}{l} - a (- b + a) & = - a \cdot (- b) - a \cdot (+ a) \\ = a b - a^{2} \end{array}

\begin{array}{l} - a (- b - a) & = - a \cdot (- b) - a \cdot (- a) \\ = a b - a^{2} \end{array}

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