Unit 5 sec 4.1 Multiplying out brackets

Unit 5 sec 4.1 Multiplying out brackets

20 December 2015

23:06

Any expression that contains brackets, such as

8a + 3b(b − 2a)

or

(2m + 3n) − (m + n − 3r),

can be rewritten without brackets. To see how to do this, let’s start by

looking at an expression that involves only numbers:

(2 + 3) × 4.

When you learned to collect like terms, you saw that 2 + 3 batches of

4 dots is the same as 2 batches of 4 dots plus 3 batches of 4 dots, as

illustrated in Figure 4. So (2 + 3) × 4 is equivalent to

2 × 4 + 3 × 4.

Here an expression containing brackets has been rewritten as an expression

without brackets:

(2 + 3) × 4 2 × 4 + 3 × 4

 (2 + 3) ×4 = 2 × 4 + 3 × 4.

It’s usual to write numbers in front of brackets, so let’s write the 4 first in

each multiplication:

4(2 + 3) = 4× 2 + 4 × 3.

Here you can see how to rewrite an expression with brackets as one

without brackets: you multiply each of the numbers inside the brackets

individually by the number outside the brackets. This is called

multiplying out the brackets, expanding the brackets, or simply

removing the brackets. The number outside the brackets is called the

multiplier. Here’s another example, with multiplier 7.

7(1 +5) = 7× 1 + 7 × 5

Multiplying out the brackets can be particularly helpful for expressions

that contain letters. The rule above applies in just the same way.

Strategy To multiply out brackets

Multiply each Remember that you must term inside the brackets by the multiplier.

multiply every term inside the

brackets, not just the first term.

Here are two examples, with multipliers a and 3, respectively.

a(b + c) = ab + ac

3(p + q2 + r) = 3p + 3q2 + 3r

It doesn’t matter whether the multiplier is before or after the brackets.

Here’s an example of multiplying out where the multiplier is after the

brackets:

(x + y)z = xz + yz.

If you prefer the multiplier to be before the brackets, then you can change

the order before multiplying out. For example,

(x + y)z = z(x + y) = zx + zy = xz + yz.

When you multiply out brackets, you often need to simplify the resultingterms, as illustrated in the next example.

Example 11 Multiplying out brackets

Multiply out the brackets in the following expression:

Tutorial clip

2a(3a + 2b).

Solution

2a(3a + 2b) = 2a × 3a + 2a × 2b

= 6a2 + 4ab

Once you’re familiar with how to multiply out brackets, it’s usually best to

simplify the terms as you multiply out, instead of first writing down an

expression containing multiplication signs. This leads to tidier expressions

and fewer errors.

For example, if you look at the expression in Example 11,

2a(3a + 2b),

you can see that when you multiply out the brackets, the first term will be

2a times 3a. You simplify this to 6a2, using the strategy of first finding the

sign, then the rest of the coefficient and then the letters, and write it down. Then you see that the second term is 2a times +2b, simplify this

to +4ab, and write it down after the first term. This gives

2a(3a + 2b) = 6a2 + 4ab.

Simplifying the terms at the same time as multiplying out is particularly

helpful when some of the terms inside the brackets, or the multiplier, have

minus signs. For example, let’s multiply out the brackets in the expression

3m(−2m + 3n − 6).

The first term is 3m times −2m, which simplifies to −6m2. Working out

the other terms in a similar way, we obtain

3m(−2m + 3n − 6) = −6m2 + 9mn − 18m.

Example 12 Multiplying out brackets involving minus signs

Multiply out the brackets in the following expression:

Tutorial clip

−a(b − a + 7).

Solution

−a(b − a + 7) = −ab + a2 − 7a

An expression containing brackets may have more than one term. For

example, the expression

x(y + 1) + 2y(y + 3)

has two terms, each containing brackets, as follows:

x(y + 1) + 2y(y + 3) .

An expression like this can be dealt with term by term, using a similar

Strategy To multiply out brackets in an expression with more than one term

1. Identify the terms. Each term after the first starts with a plus or

minus sign that isn’t inside brackets.

2. Multiply out the brackets in each term. Include the sign (plus or minus) at the start of each resulting term.

3. Collect any like terms.

Example 13 Expanding the brackets when there’s more than one term

 (a) x(y + 1) + 2y(y + 3) (b) 2r2 − r(r − s)

Solution

(a) Identify the terms. Multiply out the brackets. Then check for like

terms – there are none here.

x(y + 1) + 2y(y + 3) = xy + x + 2y2 + 6y

(b) Identify the terms. Multiply out the brackets. Collect like

terms.

2r2 − r(r − s) = 2r2 − r2 + rs

= r2 + rs

Some expressions, such as

−(a + 2b − c),

contain brackets with just a minus sign in front.

You can remove these brackets by using the fact that a minus sign in front

is just the same as multiplying by −1:

−(a + 2b − c) = −1(a + 2b − c)

= −a − 2b + c.

You can see that the overall effect is that the sign of each term in the

brackets has been changed.

An expression may also contain brackets with just a plus sign in front.

These brackets can be removed by using the fact that a plus sign in front is

just the same as multiplying by 1. For example, the expression

2x + (y − 3z)

can be simplified as follows:

2x + (y − 3z) = 2x + 1(y − 3z)

= 2x + y − 3z.

This time you can see that the effect is that all the signs in the brackets

remain as they are.

Strategy To remove brackets with a plus or minus sign in front

• If the sign is plus, keep the sign of each term inside the brackets

the same.

• If the sign is minus, change the sign of each term inside the

brackets.

Example 14 Plus and minus signs in front of brackets

Remove the brackets in the following expressions.

 (a) −(−P2 + 2Q − 3R) (b) a + (2bc − d)

Solution

(a) −(−P2 + 2Q − 3R) = +P2 − 2Q + 3R

= P2 − 2Q + 3R

(b) a + (2bc − d) = a + 2bc – d

Some expressions, such as

(x + 2)(x − 5),

contain two, or even more, pairs of brackets multiplied together.

The second form of this expression is clearly simpler than the first:

• it’s shorter and easier to understand, and

• it’s easier to evaluate for any particular value of x.

These are the attributes to aim for when you try to write an expression in

a simpler way.

However, sometimes it’s not so clear that one way of writing an expression

is better than another. For example,

x(x + 1) is equivalent to x2 + x.

Both these forms are reasonably short, and both are reasonably easy to

evaluate. So this expression doesn’t have a simplest form.

The same is true of many other expressions. One form might be better

for some purposes, and a different form might be better for other purposes.

In particular, multiplying out the brackets in an expression doesn’t always simplify it.