3.4 Multiply and Divide Integers

The topics covered in this section are:

1. Multiply integers
2. Divide integers
3. Simplify expressions with integers
4. Evaluate variable expressions with integers
5. Translate word phrases to algebraic expressions

3.4.1 Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that $a \cdot b$ means add $a, b$ times, Here, we are using the model shown below just to help us discover the pattern.

Now consider what it means to multiply $5$ by $-3$. It means subtract $5, 3$ times. Looking at subtraction as taking away, it means to take away $5, 3$ times. But there is nothing to take away, so we start by adding neutral pairs as shown in the figure below.

In both cases, we started with $15$ neutral pairs. In the case on the left, we took away $5, 3$ times and the result was $-15$. To multiply $(−5)(−3)$, we took away $−5,3$ times and the result was $15$. So we found that

$5 \cdot 3 = 15$	$-5(3)=-15$
$5(-3)=-15$	$(-5)(-3)=15$

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

MULTIPLICATION OF SIGNED NUMBERS

The sign of the product of two numbers depends on their signs.

Same signs	Product
1. Two positives 2. Two negatives	Positive Positive

Different signs	Product
1. Positive $\cdot$ negative 2. Negative $\cdot$ positive	Negative Negative

Example 1

Multiply each of the following:

1. $-9 \cdot 3$
2. $-2(-5)$
3. $4(-8)$
4. $7 \cdot 6$

When we multiply a number by $1$, the result is the same number. What happens when we multiply a number by $-1$? Let’s multiply a positive number and then a negative number by $-1$ to see what we get.

$-1 \cdot 4$	$-1(-3)$
$-4$	$3$
$-4$ is the opposite of $4$	$3$ is the opposite of $-3$

Each time we multiply a number by $-1$, we get its opposite.

Multiplication by $-1$

Multiplying a number by $-1$ gives its opposite.

$-1a = -a$

Example 2

Multiply each of the following:

1. $-1 \cdot 7$
2. $-1(-11)$

3.4.2 Divide Integers

Division is the inverse operation of multiplication. So, $15 \div 3=5$ because $5 \cdot 3=15$. In words, this expression says that $15$ can be divided into $3$ groups of $5$ each because adding five times three times gives $15$. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

$5 \cdot 3 = 15$ so $15 \div 3 = 5$	$-5(3)=-15$ so $-15 \div 3 = -5$
$(-5)(-3)=15$ so $15 \div (-3) = -5$	$5(-3)=-15$ so $-15 \div -3 = 5$

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

DIVISION OF SIGNED NUMBERS

The sign of the quotient of two numbers depends on their signs.

Same signs	Quotient
1. Two Positives 2. Two Negatives	Positive Positive

Same signs	Quotient
1. Positive & negative 2. Negative & positive	Negative Negative

Remember, you can always check the answer to a division problem by multiplying.

Example 3

Divide each of the following:

1. $-27 \div 3$
2. $-100 \div (-4)$

Just as we saw with multiplication, when we divide a number by $1$, the result is the same number. What happens when we divide a number by $-1$? Let’s divide a positive number and then a negative number by $-1$ to see what we get.

$8 \div (-1)$	$-9 \div (-1)$
$-8$	$9$
$-8$ is the opposite of $8$	$9$ is the opposite of $-9$

When we divide a number by $-1$, we get its opposite.

Division by $-1$

Dividing a number by $-1$ gives its opposite.

$a \div (-1) = -a$

Example 4

Divide each of the following:

1. $16 \div (-1)$
2. $-20 \div (-1)$

3.4.3 Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

Example 5

Simplify: $7(-2)+4(-7)-6$

Example 6

Simplify:

1. $(-2)^{4}$
2. $-2^{4}$

Example 7

Simplify: $12-3(9-12)$.

Example 8

Simplify: $8(-9) \div (-2)^{3}$.

Example 9

Simplify: $-30 \div 2 +(-3)(-7)$.

3.4.4 Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

Example 10

Evaluate $2x^{2}-3x+8$ when $x=-4$.

Example 11

Evaluate $3x+4y-6$ when $x=-1$ and $y=2$.

3.4.5 Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.

Example 12

Translate to an algebraic expression and simplify if possible: the product of $-2$ and $14$.

Example 13

Translate to an algebraic expression and simplify if possible: the quotient of $-56$ and $-7$.