Multiplying expressions

What this les­son is about

Mul­ti­ply­ing expres­sions is the step that turns a for­mula with brack­ets into one you can work with. You use it to find the area of a rec­tan­gle whose sides are writ­ten in let­ters, to open up brack­ets in an equa­tion before solv­ing it, and later to fac­torise qua­drat­ics in Class 9 and 10. The rules are the ones you already use for num­bers; only the num­bers have been replaced by let­ters.

You already know how to add and take away expres­sions. Now you will mul­ti­ply them.

An expres­sion is a piece of alge­bra made of num­bers and let­ters. $3x + 4$ is an expres­sion.

A term is one piece of an expres­sion, as Intro­duc­tion to Alge­bra sets out: a num­ber on its own, a let­ter on its own, or num­bers and let­ters joined by mul­ti­ply­ing or divid­ing. A plus or a minus sign is what sep­a­rates one term from the next. $3x + 4$ has two terms.

A fac­tor is one of the things being mul­ti­plied. In $3 \times x$, the fac­tors are $3$ and $x$.

Like terms have exactly the same let­ters. $3x$ and $2x$ are like terms. $3x$ and $3y$ are not.

You met term, fac­tor and like terms in the Early Years course Intro­duc­tion to Alge­bra. The three lines above are a reminder.

Noth­ing new is invented here. A let­ter stands for a num­ber you do not know yet. So let­ters fol­low the same mul­ti­pli­ca­tion rules as num­bers.

The prod­uct is the answer you get when you mul­ti­ply.

There are three kinds of mul­ti­pli­ca­tion to learn. Here is what each one looks like. The rest of the les­son shows why each is true.

The names for these expres­sions

The three kinds are named after the parts you mul­ti­ply. Here are the names you will need.

One mono­mial is enough for a poly­no­mial. So $3x$ is a mono­mial and a poly­no­mial at the same time.

A multi­n­o­mial only asks for terms. Its pow­ers may be any­thing at all.

$3x^{-1} + 4$ is a multi­n­o­mial. It is not a poly­no­mial, because the power $-1$ is neg­a­tive.

Remem­ber. Every poly­no­mial is a multi­n­o­mial. Not every multi­n­o­mial is a poly­no­mial.

Mul­ti­ply­ing one term by one term

$3x$ and $5y$ are mono­mi­als. Each has one term, and no power is neg­a­tive.

$3x$ really means $3 \times x$. The mul­ti­pli­ca­tion sign is hid­den, to save writ­ing.

So $3x \times 5y$ means $3 \times x \times 5 \times y$.

You may mul­ti­ply num­bers in any order. $3 \times 5$ and $5 \times 3$ both give $15$.

A let­ter stands for a num­ber, so let­ters may be moved too. Move the num­bers together, and the let­ters together.

Mul­ti­ply $3x$ by $5y$

$$\begin{aligned}&3x \times 5y \\ &= 3 \times x \times 5 \times y \\ &= 3 \times 5 \times x \times y \\ &= 15xy\end{aligned}$$

Remem­ber. Mul­ti­ply the num­bers first. Write the let­ters after the num­ber.

When the same let­ter is in both parts

$x^{2}$ means $x \times x$. $x^{3}$ means $x \times x \times x$.

The small raised num­ber is called the power. It tells you how many of that let­ter are mul­ti­plied together.

Mul­ti­ply $x^{2}$ by $x^{3}$ and you have two $x$, and then three more $x$.

That is five $x$ mul­ti­plied together. $2 + 3 = 5$, so $x^{2} \times x^{3} = x^{5}$.

You add the pow­ers because you are count­ing fac­tors. You are not mul­ti­ply­ing the pow­ers.

A let­ter with no power writ­ten has power $1$. So $y$ means $y^{1}$.

Mul­ti­ply $4x^{4}$ by $3x^{2}$

$$\begin{aligned}&4x^{4} \times 3x^{2} \\ &= 4 \times 3 \times x^{4} \times x^{2} \\ &= 12 \times x^{6} \\ &= 12x^{6}\end{aligned}$$

Remem­ber. Add the pow­ers of a repeated let­ter. $4 + 2 = 6$, so $x^{4} \times x^{2} = x^{6}$.

A mono­mial with a minus sign

Signs fol­low the num­ber rules: a neg­a­tive times a pos­i­tive is neg­a­tive. Mul­ti­ply $-2xy$ by $7x^{2}$.

$$\begin{aligned}&-2xy \times 7x^{2} \\ &= (-2 \times 7) \times x \times x^{2} \times y \\ &= -14x^{3}y\end{aligned}$$

The $x$ has power $1$, so $x \times x^{2} = x^{1 + 2} = x^{3}$. The $y$ appears only once, so it is sim­ply writ­ten after the $x^{3}$.

Mul­ti­ply­ing one term by a whole expres­sion

Now the sec­ond part has more than one term. $3x + 4$ has two terms, so it is a bino­mial.

Test the rule with num­bers first. $3(4 + 5)$ is $3 \times 9$, which is $27$.

Now share the $3$ out. $3 \times 4 + 3 \times 5 = 12 + 15 = 27$. The two answers match.

This rule is writ­ten as $a(b + c) = ab + ac$.

It works because $a$ lots of $b + c$ means $a$ lots of $b$, plus $a$ lots of $c$.

Mul­ti­ply $2x$ by $3x + 4$

$$\begin{aligned}&2x(3x + 4) \\ &= 2x \times 3x + 2x \times 4 \\ &= 6x^{2} + 8x\end{aligned}$$

$2x \times 3x$ gives $6x^{2}$. The pow­ers of $x$ add, and $1 + 1 = 2$.

Remem­ber. Every term inside the bracket must be mul­ti­plied. Miss one and the answer is wrong.

One term by a three-term expres­sion

The rule stretches to any num­ber of terms in the bracket. Mul­ti­ply $3x$ by $2x^{2} - 5x + 4$.

$$\begin{aligned}&3x(2x^{2} - 5x + 4) \\ &= 3x \times 2x^{2} + 3x \times (-5x) + 3x \times 4 \\ &= 6x^{3} - 15x^{2} + 12x\end{aligned}$$

Three terms inside the bracket give three terms in the answer. Keep each sign with the term that fol­lows it.

Mul­ti­ply­ing a whole expres­sion by a whole expres­sion

Now both parts have more than one term. Every term of the first must be mul­ti­plied by every term of the sec­ond.

This is where marks are lost. One pair gets missed.

A grid stops that. Put one expres­sion across the top, and one down the side.

Two terms across the top and two down the side make $2 \times 2 = 4$ boxes. Each box holds one prod­uct, so noth­ing can hide.

Find the prod­uct $(x + 2)(x + 3)$

$$\begin{aligned}&(x + 2)(x + 3) \\ &= x^{2} + 3x + 2x + 6 \\ &= x^{2} + 5x + 6\end{aligned}$$

$3x$ and $2x$ are like terms. $3 + 2 = 5$, so together they make $5x$.

The answer has three terms, so it is a tri­no­mial.

A prod­uct with a minus sign

Find $(2x + 3)(x - 4)$. Treat $-4$ as a term with its sign. The four prod­ucts are $2x \times x = 2x^{2}$, $2x \times (-4) = -8x$, $3 \times x = 3x$ and $3 \times (-4) = -12$.

$$\begin{aligned}&(2x + 3)(x - 4) \\ &= 2x^{2} - 8x + 3x - 12 \\ &= 2x^{2} - 5x - 12\end{aligned}$$

The like terms $-8x$ and $3x$ com­bine to $-5x$, because $-8 + 3 = -5$.

Two worked sim­pli­fi­ca­tions

Open the brack­ets first. Then col­lect the like terms.

Sim­plify $3a(b + c) + ab - 3ac$

$$\begin{aligned}&3a(b + c) + ab - 3ac \\ &= 3ab + 3ac + ab - 3ac \\ &= 3ab + ab + 3ac - 3ac \\ &= 4ab\end{aligned}$$

$3ac$ and $-3ac$ are like terms with oppo­site signs. Take $3ac$ away from $3ac$ and noth­ing is left. So they dis­ap­pear.

$3ab$ and $ab$ are like terms as well. Together they make $4ab$.

Sim­plify $a - 4b + 3(a + c)$

$$\begin{aligned}&a - 4b + 3(a + c) \\ &= a - 4b + 3a + 3c \\ &= 4a - 4b + 3c\end{aligned}$$

$a$ and $3a$ are like terms, so they make $4a$. $-4b$ and $3c$ have no part­ner, so they stay.

The order of work

1. Mul­ti­ply every term by every term.
2. Mul­ti­ply the num­bers first.
3. Then write the let­ters after the num­ber.
4. Add the pow­ers of any let­ter that appears twice.
5. Col­lect the like terms at the end.

Remem­ber. Col­lect like terms last. Do all the mul­ti­ply­ing first.

Prac­tice

Now try three longer ones. Open the brack­ets, then col­lect.

1. Sim­plify $2a(b + c) + 2ab - 2ac$
2. Sim­plify $2a - 3b + 4(a + b)$
3. Find the prod­uct $(x + 5)(x + 2)$

Com­mon mis­takes

• Mul­ti­ply­ing the pow­ers instead of adding them: $x^{2} \times x^{3}$ is $x^{5}$, not $x^{6}$.
• Mul­ti­ply­ing only the first term in the bracket: $5(x + 2)$ is $5x + 10$, not $5x + 2$.
• Miss­ing a pair when both parts have two terms. Use the grid so that all four prod­ucts are writ­ten down.
• Los­ing a minus sign. In $(2x + 3)(x - 4)$, the prod­uct $3 \times (-4)$ is $-12$.
• Adding unlike terms: $x^{2}$ and $5x$ can­not be com­bined, because the let­ter parts dif­fer.
• Col­lect­ing like terms before all the mul­ti­ply­ing is fin­ished.

Key terms

Term

A num­ber, a let­ter, or num­bers and let­ters joined by mul­ti­ply­ing or divid­ing; terms are sep­a­rated by plus and minus signs.

Like terms

Terms with exactly the same let­ters raised to the same pow­ers, such as $3x$ and $2x$.

Power

The small raised num­ber that tells how many times a let­ter is mul­ti­plied by itself.

Prod­uct

The answer to a mul­ti­pli­ca­tion.

Mono­mial

An expres­sion of one term in which every power of a let­ter is a whole num­ber.

Bino­mial

An expres­sion of two mono­mi­als, such as $x + 2$.

Dis­trib­u­tive law

The rule $a(b + c) = ab + ac$: a fac­tor out­side a bracket mul­ti­plies every term inside.

Answers

Prac­tice

1. $3x \times 5y = 15xy$
2. $4a \times 6b = 24ab$
3. $x^{3} \times x^{4} = x^{3 + 4} = x^{7}$
4. $2a^{2} \times 5a^{3} = 10a^{5}$
5. $5(x + 2) = 5x + 10$
6. $3a(2a + b) = 6a^{2} + 3ab$
7. $(x + 1)(x + 4) = x^{2} + 4x + x + 4 = x^{2} + 5x + 4$
8. $(a + 3)(a + 2) = a^{2} + 2a + 3a + 6 = a^{2} + 5a + 6$
9. $2x(3x + 4y) = 6x^{2} + 8xy$

The three longer ones

1. $2a(b + c) + 2ab - 2ac = 2ab + 2ac + 2ab - 2ac = 4ab$
2. $2a - 3b + 4(a + b) = 2a - 3b + 4a + 4b = 6a + b$
3. $(x + 5)(x + 2) = x^{2} + 2x + 5x + 10 = x^{2} + 7x + 10$