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.
