Algebra · Algebraic Expressions · Lesson 3

Multiplying and Dividing Algebraic Expressions

Carpenters, tilers and farmers multiply lengths to find areas, and divide to share land or materials equally. In this lesson you will multiply and divide algebraic terms — numbers with numbers, letters with letters — including negative and fraction coefficients, and use this to write the area of shapes.

NaCCA B7.2.2.1.3 ⏱ about 45 min

By the end you will be able to

✔Multiply algebraic terms by multiplying the coefficients and then the letters, e.g. $3 x \times 2 y = 6 x y$ .
✔Use the sign rules to multiply and divide terms with negative coefficients.
✔Write $x \times x$ as $x^{2}$ and $x \times x \times x$ as $x^{3}$ .
✔Divide algebraic terms by writing them as a fraction and cancelling common factors.
✔Write expressions for the area of rectangles, squares and triangles with algebraic sides.

Key words

Product: The result of multiplying. The product of $3 x$ and $2 y$ is $6 x y$ .
Quotient: The result of dividing. The quotient of $12 x y$ and $4 x$ is $3 y$ .
Index (power): A small raised number showing repeated multiplication: $x^{2} = x \times x$ .
Cancel: Divide the top and bottom of a fraction by the same factor, e.g. $\frac{x}{x} = 1$ .

Curriculum reference

Content standard: B7.2.2.1 Simplify algebraic expressions involving the four basic operations and substituting values to evaluate algebraic expressions.

Indicator: Perform multiplication and division of algebraic expressions with rational coefficients.

Starter

Tiling a classroom floor in Cape Coast

A contractor is tiling a classroom floor at a school in Cape Coast. He does not yet know the exact size of the tiles he will buy, so he calls the length of one tile x and the width y. The floor is 3 tiles long and 2 tiles wide. How much floor area is that? And if he has to share the whole area equally among the tiles, how many does he need? You will answer both questions with one multiplication and one division of algebraic terms.

1. Multiplying terms: numbers first, then letters

A term like $3 x$ already hides a multiplication: it means $3 \times x$ . So when we multiply two terms we are really multiplying a list of numbers and letters together. Because the order of a multiplication does not matter, we can multiply the numbers together and the letters together.

1Multiply the coefficients (the numbers).
2Multiply the letters, writing them next to each other in alphabetical order.
3If the same letter appears more than once, write it as a power: $x \times x = x^{2}$ .

Multiplication	Numbers	Letters	Answer
$3 \times 4 x$	3 × 4 = 12	$x$	$12 x$
$2 a \times 5 b$	2 × 5 = 10	$a \times b = ab$	$10 ab$
$4 x \times 8 x$	4 × 8 = 32	$x \times x = x^{2}$	$32 x^{2}$
$\frac{1}{2} x \times 4 y$	$\frac{1}{2} \times 4 = 2$	$x y$	$2 x y$
$a \times a \times a$	1	$a^{3}$	$a^{3}$

▶ The rectangle 3x by 2y is made of 6 small pieces, each of area xy, so its area is 6xy.

2. Signs: multiplying with negative numbers

Coefficients can be negative. Work out the sign first, then multiply as usual. The sign rules are the same for multiplication and division:

Signs	Result	Example
+ × +	+	$4 \times 5 y = 20 y$
− × + or + × −	−	$- 3 \times 5 x = - 15 x$
− × −	+	$- 6 p \times - 2 q = 12 pq$

With three or more terms, multiply two at a time. For $- 2 a \times 4 b \times 5 c$ : first $- 2 a \times 4 b = - 8 ab$ , then $- 8 ab \times 5 c = - 40 ab c$ . A quick check: count the negative signs — an odd number of negatives gives a negative answer, an even number gives a positive answer.

3. Dividing terms by cancelling

To divide algebraic terms, write the division as a fraction. Then divide the numbers, and cancel any letter that appears both on top and underneath, because anything divided by itself is 1 ( $\frac{x}{x} = 1$ ).

▶ Cancel the letter that is on top and underneath, then divide the numbers. 12xy ÷ 4x = 3y.

$\frac{18 x ^{2}}{6 x} = \frac{18 \times x \times x}{6 \times x} = 3 x$
$\frac{- 30 ab}{6 b} = - 5 a$ (negative ÷ positive = negative)
$\frac{12 x y}{16 x} = \frac{3 y}{4}$ (12 and 16 share a factor of 4, so the number part stays a fraction)
$10 x \div 4 = \frac{10 x}{4} = \frac{5 x}{2}$

4. Areas with algebraic sides

Multiplying terms lets us write the area of a shape whose sides are given with letters. Use the usual area formulas and simplify.

Shape	Sides	Area
Rectangle	length $4 a$ , width $3 b$	$4 a \times 3 b = 12 ab$
Square	side $5 x$	$5 x \times 5 x = 25 x^{2}$
Triangle	base $6 x$ , height $4 y$	$\frac{1}{2} \times 6 x \times 4 y = 12 x y$

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Multiplying two terms

Simplify

4 a \times 8 a

Show the solution

Step 1Numbers: 4 × 8 = 32.
Step 2Letters: $a \times a = a^{2}$ .
Step 3Answer: $32 a^{2}$ .

Answer: $32 a^{2}$

Example 2

Three terms with a negative

Simplify

- 2 a \times 4 b \times 5 c

Show the solution

Step 1Sign: one negative, so the answer is negative.
Step 2Numbers: 2 × 4 × 5 = 40.
Step 3Letters: $ab c$ .
Step 4Answer: $- 40 ab c$ .

Answer: $- 40 ab c$

Example 3

Dividing terms

Simplify

\frac{- 30 ab}{6 b}

Show the solution

Step 1Sign: negative ÷ positive = negative.
Step 2Numbers: 30 ÷ 6 = 5.
Step 3Letters: $\frac{ab}{b} = a$ (the $b$ cancels).
Step 4Answer: $- 5 a$ .

Answer: $- 5 a$

Example 4

Fraction coefficients

Simplify (a)

\frac{1}{2} x \times 4 y

(b)

\frac{12 x y}{16 x}

Show the solution

Step 1(a) $\frac{1}{2} \times 4 = 2$ , letters $x y$ : answer $2 x y$ .
Step 2(b) Cancel $x$ : $\frac{12 y}{16}$ .
Step 3Simplify $\frac{12}{16} = \frac{3}{4}$ : answer $\frac{3 y}{4}$ .

Answer: (a) $2 x y$ (b) $\frac{3 y}{4}$

Example 5

Area of a triangle

A triangular flower bed has base

6 x

m and height

4 y

m. Write an expression for its area.

Show the solution

Step 1Area of triangle $= \frac{1}{2} \times$ base $\times$ height.
Step 2 $= \frac{1}{2} \times 6 x \times 4 y$ .
Step 3Numbers: $\frac{1}{2} \times 6 \times 4 = 12$ . Letters: $x y$ .
Step 4Area $= 12 x y$ m².

Answer: $12 x y$ m²

Summary

To multiply terms, multiply the numbers, then the letters: $2 a \times 5 b = 10 ab$ .
The same letter multiplied by itself gives a power: $x \times x = x^{2}$ , so $4 x \times 8 x = 32 x^{2}$ .
Signs: same signs give +, different signs give −.
To divide, write a fraction, divide the numbers and cancel letters found on top and underneath: $\frac{12 x y}{4 x} = 3 y$ .
Multiplying is not adding: $2 a \times 3 b = 6 ab$ , not $5 ab$ .

Tiling a classroom floor in Cape Coast

1. Multiplying terms: numbers first, then letters

2. Signs: multiplying with negative numbers

3. Dividing terms by cancelling

4. Areas with algebraic sides

Worked examples

Multiplying two terms

Three terms with a negative

Dividing terms

Fraction coefficients

Area of a triangle

Summary

Practice: 15 questions