Multiply and Divide Integers

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.

This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.

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.

This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives.  Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.

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 signsProduct
1. Two positives
2. Two negatives
Positive
Positive
Different signsProduct
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. $7 \cdot 6$
Solution

Part 1.$-9 \cdot 3$
Multiply, noting that the signs are different and so the product is negative.$-27$
Part 2.$-2(-5)$
Multiply, noting that the signs are the same and so the product is positive.$10$
Part 3.$4(-8)$
Multiply, noting that the signs are different and so the product is negative.$-32$
Part 4.$7 \cdot 6$
The signs are the same, so the product is positive.$42$

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. $-1(-11)$
Solution

Part 1.
The signs are different, so the product will be negative.$-1 \cdot 7$
Notice that $-7$ is the opposite of $7$.$-7$
Part 2.
The signs are the same, so the product will be positive.$-1(-11)$
Notice that $11$ is the opposite of $-11$.$11$

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 signsQuotient
1. Two Positives
2. Two Negatives
Positive
Positive
Same signsQuotient
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. $-100 \div (-4)$
Solution

Part 1.$-27 \div 3$
Divide, noting that the signs are different and so the quotient is negative.$-9$
Part 2.$-100 \div (-4)$
Divide, noting that the signs are the same and so the quotient is positive.$25$

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. $-20 \div (-1)$
Solution

Part 1.$16 \div (-1)$
The dividend, $16$, is being divided by $-1$.$-16$
Dividing a number by $-1$ gives its opposite.
Notice that the signs were different, so the result was negative.
Part 2.$-20 \div (-1)$
The dividend, $-20$, is being divided by $-1$.$20$
Dividing a number by $-1$ gives its opposite.

Notice that the signs were the same, so the quotient was positive.

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

Solution

We use the order of operations. Multiply first and then add and subtract from left to right.

$7(-2)+4(-7)-6$
Multiply first.$-14+(-28)-6$
Add.$-42-6$
Subtract.$-48$

Example 6

Simplify:

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

The exponent tells how many times to multiply the base.

Part 1. The exponent is $4$ and the base is $-2$. We raise $-2$ to the fourth power.

$(-2)^{4}$
Write in expanded for,.$(-2)(-2)(-2)(-2)$
Multiply.$4(-2)(-2)$
Multiply.$-8(-2)$
Multiply.$16$

Part 2. The exponent is $4$ and the base is $2$. We raise $2$ to the fourth power and then take the opposite.

$-2^{4}$
Write in expanded for,.$-(2 \cdot 2 \cdot 2 \cdot 2)$
Multiply.$-(4 \cdot 2 \cdot 2)$
Multiply.$-(8 \cdot 2)$
Multiply.$-16$

Example 7

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

Solution

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

$12-3(9-12)$
Subtract the parentheses first.$12-3(-3)$
Multiply.$12-(-9)$
Subtract.$21$

Example 8

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

Solution

We simplify the exponent first, then multiply and divide.

$8(-9) \div (-2)^{3}$
Simplify the exponent.$8(-9) \div (-8)$
Multiply.$-72 \div (-8)$
Divide.$9$

Example 9

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

Solution

First we will multiply and divide from left to right. Then we will add.

$-30 \div 2+(-3)(-7)$
Divide.$-15+(-3)(-7)$
Multiply.$-15+21$
Add.$6$

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 $x=-4$.

Solution

$2x^{2}-3x+8$
Substitute $-4$ for $x$.$2(-4)^{2}-3(-4)+8$
Simplify exponents.$2(16)-3(-4)+8$
Multiply.$32-(-12)+8$
Subtract.$44+8$
Add.$52$

Keep in mind that when we substitute $-4$ for $x$, we use parentheses to show the multiplication. Without parentheses, it would look like $2 \cdot -4^{2} -3 \cdot -4 +8$.

Example 11

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

Solution

$3x+4y-6$
Substitute $x=-1$ and $y=2$.$3(-1)+4(2)-6$
Multiply.$-3+8-6$
Simplify.$-1$

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 $14$.

Solution

The word product tells us to multiply.

the product of $-2$ and $14$
Translate.$(-2)(14)$
Simplify.$-28$

Example 13

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

Solution

The word quotient tells us to divide.

the quotient of $-56$ and $-7$
Translate.$-56 \div (-7)$
Simplify.$8$
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