Expanding and Factoring Linear Expressions | Grade 7 Math | LevelKite

Here is a quick way to find $3 \times 14$ . Split $14$ into $10 + 4$ and multiply each part by $3$ :

$3(10 + 4) = 3 \cdot 10 + 3 \cdot 4 = 30 + 12 = 42$

That matches $3 \times 14 = 42$ . The same amount is written first as a product (a multiplication) and then as a sum (an addition).

An expression is a mathematical phrase built from numbers, letters and operations. You have already used properties of operations to rewrite expressions without changing their value. In this lesson you will use one of those properties to switch between a product and a sum, even when one of the numbers is unknown.

Let $x$ be a variable, a letter that stands for any number. Because $3$ is a whole number, $3(x + 4)$ means three copies of $x + 4$ added together. You can add numbers in any order and in any grouping, so you can collect the three copies of $x$ and the three copies of $4$ :

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

So $3(x + 4)$ and $3x + 12$ have the same value, whatever number $x$ stands for. The equals signs in that chain do not ask you to find $x$ . They say that the expressions have the same value for every value of $x$ .

A few words will help you talk about expressions like these:

Terms are the parts of an expression that are added. Read a subtraction as adding the opposite, so the terms of $2x - 5$ are $2x$ and $-5$ .
A coefficient is the number that multiplies a variable. In $2x$ , the coefficient is $2$ .
A constant is a term with no variable, such as $-5$ .
A factor is a number or expression that is multiplied. In $3(x + 4)$ , the factors are $3$ and $x + 4$ .
A linear expression can be written as a sum of terms that are each a constant or a coefficient times a variable. The variable is never squared or multiplied by another variable.
Two expressions are equivalent only if they are equal for every value of the variable.

Key concept

The distributive property works in both directions

The distributive property says that multiplying a sum by a number gives the same result as multiplying each term of the sum by that number and then adding the products. It is true for all numbers $a$ , $b$ and $c$ , including negative numbers, fractions and decimals.

Read it left to right to expand: a product becomes a sum.
Read it right to left to factor: a sum becomes a product.

Because the property is true for all numbers, the product form and the sum form are equal for every value of the variable. That is what makes them equivalent.

a(b + c) = ab + ac

To expand a product such as $5(2x - 3)$ , multiply the factor outside the parentheses by every term inside. The sum you get is called the expanded form.

Subtracting $3$ is the same as adding its opposite, $-3$ , so the inside is the sum $2x + (-3)$ . Now the distributive property applies, with $a = 5$ , $b = 2x$ and $c = -3$ :

$5(2x - 3) = 5 \cdot 2x + 5 \cdot (-3) = 10x + (-15) = 10x - 15$

Two facts help you find each product:

To multiply a number by a term such as $2x$ , multiply it by the coefficient: $5 \cdot 2x = (5 \cdot 2)x = 10x$ . Changing the grouping of factors does not change a product.
For two numbers that are not zero, the product is positive when their signs are the same and negative when their signs are different. Here $5 \cdot (-3) = -15$ .

Each step uses a property that is true for all numbers, so $10x - 15$ equals $5(2x - 3)$ for every value of $x$ . The two expressions are equivalent.

The outside factor can be a whole number, a fraction or a decimal

Product	Multiply every term	Expanded form
$7(x + 2)$	$7 \cdot x + 7 \cdot 2$	$7x + 14$
$\frac{2}{3}(9x - 6)$	$\frac{2}{3} \cdot 9x + \frac{2}{3} \cdot (-6)$	$\frac{18}{3}x + \frac{-12}{3} = 6x + (-4) = 6x - 4$
$1.5(4x + 6)$	$1.5 \cdot 4x + 1.5 \cdot 6$	$6x + 9$

Worked example

Expand with a negative factor

Expand $-4(2x - 5)$ , where $x$ stands for any number.

Rewrite the subtraction inside the parentheses as adding the opposite. Subtracting 5 gives the same result as adding negative 5, so the value does not change.
$-4(2x - 5) = -4(2x + (-5))$
Use the distributive property: multiply the outside factor, negative 4, by every term inside the parentheses. The property is true for all numbers, so this sum equals the product for every value of x.
$-4(2x + (-5)) = (-4)(2x) + (-4)(-5)$
Find the first product. Group negative 4 with the coefficient 2, which does not change the product. The two numbers have different signs, so the result is negative.
$(-4)(2x) = (-4 \cdot 2)x = -8x$
Find the second product. Both numbers are negative, so their signs are the same and the product is positive.
$(-4)(-5) = 20$
Write the two products as a sum. This is the expanded form.
$-4(2x - 5) = -8x + 20$

Answer

The expanded form of $-4(2x - 5)$ is $-8x + 20$ . The two expressions are equivalent because they are equal for every value of $x$ .

Testing a correct and a mistaken expansion at four values of x

Value of x	Original expression	Correct expansion	Mistaken expansion
$x$	$-4(2x - 5)$	$-8x + 20$	$-8x - 20$
$0$	$-4(0 - 5) = -4(-5) = 20$	$0 + 20 = 20$	$0 - 20 = -20$
$1$	$-4(2 - 5) = -4(-3) = 12$	$-8 + 20 = 12$	$-8 - 20 = -28$
$-2$	$-4(-4 - 5) = -4(-9) = 36$	$16 + 20 = 36$	$16 - 20 = -4$
$\frac{1}{2}$	$-4(1 - 5) = -4(-4) = 16$	$-4 + 20 = 16$	$-4 - 20 = -24$

To factor an expression, run the distributive property backwards: start with a sum and write it as a product, called the factored form.

Look at $6x + 15$ . Both terms can be written with a factor of $3$ , because $6x = 3 \cdot 2x$ and $15 = 3 \cdot 5$ . A number that is a factor of every term is a common factor. So:

$6x + 15 = 3 \cdot 2x + 3 \cdot 5 = 3(2x + 5)$

When the coefficient and the constant are integers (whole numbers and their opposites), three moves do the job:

Find the greatest common factor (GCF): the largest whole number that divides the coefficient and the constant with no remainder.
Divide each term by the GCF. The GCF is never zero, so you can always divide by it.
Write the GCF outside the parentheses and the results of the divisions, called quotients, inside.

Keep the factor outside. On its own, $2x + 5$ is not equivalent to $6x + 15$ : at $x = 0$ one equals $5$ and the other equals $15$ . Multiplying by $3$ undoes the division by $3$ , so $3(2x + 5)$ equals $6x + 15$ for every value of $x$ .

Worked example

Factor out the greatest common factor

Factor $12n + 18$ , where $n$ stands for any number.

Find the greatest common factor of the coefficient 12 and the constant 18. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 18 are 1, 2, 3, 6, 9 and 18. The largest number on both lists is 6.
$\text{GCF of } 12 \text{ and } 18 = 6$
Divide each term by the greatest common factor to find what goes inside the parentheses. You are dividing by 6, which is not zero, so the division is allowed.
$12n \div 6 = 2n \qquad 18 \div 6 = 3$
Rewrite each term as 6 times its quotient. The value does not change, because 6 times 2n is 12n and 6 times 3 is 18.
$12n + 18 = 6 \cdot 2n + 6 \cdot 3$
Use the distributive property from right to left to write the sum as a product. The property is true for all numbers, so the product equals the sum for every value of n.
$6 \cdot 2n + 6 \cdot 3 = 6(2n + 3)$
Check by expanding your answer with the distributive property. You get the original expression back.
$6(2n + 3) = 6 \cdot 2n + 6 \cdot 3 = 12n + 18$

Answer

The factored form of $12n + 18$ is $6(2n + 3)$ , which equals $12n + 18$ for every value of $n$ . It is fully factored because $2$ and $3$ share no factor greater than $1$ . A smaller factor, as in $2(6n + 9)$ , is equivalent but leaves a common factor of $3$ inside.

Worked example

Factor out a negative number

Factor $-15y + 10$ fully by taking out a negative number, so that the coefficient of $y$ inside the parentheses is positive. Here $y$ stands for any number.

The greatest common factor of 15 and 10 is 5. Factoring out 5 would leave negative 3y inside the parentheses. To make that coefficient positive, factor out negative 5 instead.
$\text{GCF of } 15 \text{ and } 10 = 5$
Divide each term by negative 5, which is not zero. Negative 15 and negative 5 have the same sign, so the first quotient is positive. Positive 10 and negative 5 have different signs, so the second quotient is negative.
$-15y \div (-5) = 3y \qquad 10 \div (-5) = -2$
Write negative 5 outside the parentheses and the two quotients inside. Adding negative 2 is the same as subtracting 2.
$-15y + 10 = -5(3y + (-2)) = -5(3y - 2)$
Check by expanding with the distributive property. Negative 5 multiplies both terms, and you get the original expression back.
$-5(3y - 2) = (-5)(3y) + (-5)(-2) = -15y + 10$

Answer

The factored form is $-5(3y - 2)$ , which equals $-15y + 10$ for every value of $y$ . It is fully factored because $3$ and $2$ share no factor greater than $1$ . The form $5(-3y + 2)$ is also equivalent, but its inside coefficient is negative.

The big idea: expanding and factoring are one property read in opposite directions.

To expand, multiply the outside factor by every term inside the parentheses, and take care with the signs.
To factor, find a common factor of every term, divide each term by it (a common factor is never zero), and write it outside the parentheses.

Neither move changes the value of the expression. The product form and the sum form are equal for every value of the variable, and that is exactly what equivalent means.

Next, you will use these skills to decide whether two linear expressions are equivalent and to find errors in simplifying and expanding.