# 1. Introduction

Quantities are enclosed within brackets to indicate that they are to be treated as a single entity. If we wish to subtract, say, 3a − 2b from 4a − 5b then we do this as follows.

Example 1:

(a) Simplify (4a − 5b) − (3a − 2b)

Solution:

 (4a − 5b) − (3a − 2b) = 4a − 5b − 3a− (−2b) = 4a − 5b − 3a + 2b = 4a − 3a − 5b + 2b = a − 3b
(b)Simplify (7x + 5y) − (2x − 3y)

Solution:

 (7x + 5y) − (2x − 3y) = 7x + 5y − 2x− (−3y) = 7x + 5y − 2x + 3y = 7x − 2x + 5y + 3y = 5x + 8y

When there is more than one bracket it is usually necessary to begin with the inside bracket and work outwards:

Example 2: Simplify the following expressions by removing the brackets.

(a)3a c + (5a − 2b − [3ac + 2b])

Solution: We have:

 3a − c+(5a − 2b − [3a − c + 2b]) = 3a − c + (5a − 2b − 3a + c − 2b) = 3a − c + (2a − 4b + c) = 3a − c + 2a − 4b + c = 3a + 2a − 4b − c + c = 5a − 4b
(b)−{3y − (2x − 3y) + (3x − 2y)} + 2x

Solution: Similarly, we have:

 −{3y − (2x − 3y) + (3x − 2y)} + 2x = −{3y − 2x + 3y + 3x − 2y} + 2x = −{3y + 3y − 2y + 3x − 2x} + 2x = −{4y + x} + 2x = −4y − x + 2x = x − 4y

Exercise 1: Remove the brackets from each of the following expressions and simplify as far as possible.

(a) x − (yz) + x + (yz) − (z + x)

Solution:

 x − (y − z) + x + (y − z) − (z + x) = x − y + z + x + y − z − z − x = x + x − x − y + y + z − z − z = x − z
(b)2x − (5y + [3zx]) − (5x − [y + z])

Solution:

 2x − (5y + [3z − x]) − (5x − [y + z]) = 2x − (5y + 3z − x) − (5x − y − z) = 2x − 5y − 3z + x − 5x + y + z = 2x + x − 5x − 5y + y − 3z + z = −2x − 4y − 2z
(c) 3/a + b + 7/a − 2b

Solution:

 3/a + b + 7/a − 2b = 3/a + 7/a + b − 2b = 3 + 7/a − b = 10/a − b
(d)a − (b + [c − {ab}])

Solution:

 a − (b + [c − {a − b}])c = a − (b + [c − a + b]) = a − (b + c − a + b) = a − (2b + c − a) = a − 2b − c + a = 2a − 2b − c

Click on questions to reveal their solutions

# 2. Distributive Rule

A quantity outside a bracket multiplies each of the terms inside the bracket. This is known as the distributive rule.

Example 3:

(a)3(x − 2y) => 3x − 6y

(b)2x(x − 2y + z) => 2x2 − 4xy + 2xz

(c)7y − 4(2x − 3) => 7y − 8x + 12

This is a relatively simple rule but, as in all mathematical arguments, a great deal of care must be taken to apply it correctly.

Exercise 2: Remove the brackets and simplify the following expressions:

(a)5x − 7x2 − (2x)2

Solution: First note that (2x)2 = (2x) × (2x) = 4x2

 5x − 7x2 − (2x)2 = 5x − 7x2 − 4x2 = 5x − 11x2
(b) (3y)2+x2−(2y2)

Solution:

 (3y)2 + x2 − (2y)2 = 9y2 + x2 − 4y2 = 9y2 − 4y2 + x2 = 5y2 + x2
(c) 3a + 2(a + 1)

Solution:

 3a + 2(a + 1) = 3a + 2a + 2 = 5a + 2
(d) 5x − 2x(x − 1)

Solution:

 5x − 2x(x − 1) = 5x − 2x2 + 2x = 7x − 2x2
(e) 3xy − 2x(y − 2)

Solution:

 3xy − 2x(y − 2) = 3xy − 2xy + 4x = xy + 4x
(f) 3a(a − 4) − a(a − 2)

Solution:

 3a(a − 4) − a (a − 2) = 3a2 − 12a − a2 + 2a = 3a2 − a2 + 2a − 12a = 2a2 − 10a

Click on questions to reveal their solution

In the case of two brackets being multiplied together, to simplify the expression first choose one bracket as a single entity and multiply this into the other bracket.

Example 4: For each of the following expressions, multiply out the brackets and simplify as far as possible.

(a) (x + 5)(x + 2)

Solution:

 (x + 5)(x + 2) = (x + 5) x + (x + 5) 2 = x (x + 5) + 2 (x + 5) = x2 + 5x + 2x + 10 = x2 + 7x + 10
(b) (3x − 2)(2y + 3)

Solution:

 (3x − 2)(2y + 3) = (3x − 2) 2y + (3x−2) 3 = 2y (3x − 2) + 3 (3x − 2) = 6xy − 4y + 9x − 6

Try this short quiz:

Quiz 1: To which of the following does the expression (2x − 1)(x + 4) simplify?

(a)2x2 − 2x + 4Incorrect - please try again!
(b)2x2 − 7x + 4Incorrect - please try again!
(c)2x2 + 7x − 4Correct - well done!
(d)2x2 + 2x − 4Incorrect - please try again!

Explanation:

 (2x − 1)(x + 4) = (2x − 1)x + (2x − 1)4 = (2x2 − x) + (8x − 4) = 2x2 − x + 8x − 4 = 2x2 + 7x − 4

# 3. FOIL

When it comes to expanding a bracketed expression like (a+c)(x+y) there is a simple way to remember all of the terms. This is the word FOIL, and stands for:

Take products of the
First Outside Inside Last

This is illustrated in the following:

 F O I L (a+c)(x+y) = ax + ay + cx + cy

These terms are the products of the pairs highlighted below:

 F O I L (a+c)(x+y), (a+c)(x+y), (a+c)(x+y), (a+c)(x+y)

There are two special cases of brackets that are worth remembering:

(x+y)2, which is a complete square, and
(x+y)(xy), which is a difference of two squares.

These appear in the following exercises:

Exercise 3: Remove the brackets from each of the following expressions using FOIL.

(a)(x + y)2

Solution:

 (x + y)2 = (x + y)(x + y) = x2 + xy + yx + y2 using FOIL = x2 + 2xy + y2

This is an IMPORTANT result and should be committed to memory. Here x is the first member of the the bracket and y is the second. The rule for the square of (x + y), i.e. (x + y)2 is:

 x2 + 2xy + y2 (square the first) + (twice the product) + (square the last)
(b)(x + y)(xy)

Solution: Using FOIL again:

 (x + y)(x − y) = x2 + xy − xy + y2 = x2 − y2

The form of the solution is the reason for the name difference of two squares. This is another important result that is worth committing to memory.

(c)(x + 4)(x + 5)

Solution: Using FOIL:

 (x + 4)(x + 5) = x2 + 5x + 4x + 20 = x2 + 9x + 20
(d)(y + 1)(y + 3)

Solution: Using FOIL:

 (y + 1)(y + 3) = y2 + 3y + y + 3 = y2 + 4y + 3
(e)(2y + 1)(y − 3)

Solution: Using FOIL:

 (2y + 1)(y − 3) = 2y2 − 6y + y − 3 = 2y2 − 5y − 3
(f)2(x − 3)2 − 3(x + 1)2

Solution: This one is best done in parts. First we have:

(x − 3)2 = x2 − 6x + 9

and:

(x + 1)2 = x2 + 2x + 1

Thus:

 2(x − 3)2 − 3(x + 1)2 = 2(x2 − 6x + 9) − 3(x2 + 2x + 1) = 2x2 − 12x + 18 − 3x2 − 6x − 3 = 2x2 − 3x2 − 12x − 6x + 18 − 3 = −x2 − 18x + 15

Click on questions to reveal their solutions

Quiz 2: To which of the following expressions does 9 − (x − 3)2 simplify?

(a)x2Incorrect - please try again!
(b)6xx2Correct - well done!
(c)18 − x2Incorrect - please try again!
(d)6x + x2Incorrect - please try again!

Explanation: First note that (x − 3)2 = x2 − 6x + 9, so:

 9 − (x − 3)2 = 9 − (x2 − 6x + 9) = 9 − x2 + 6x − 9 = 9 − 9 − x2 + 6x = −x2 + 6x = 6x − x2

# 4. Quiz on Brackets

In each of the following, remove the brackets, simplify the expression, and choose the solution from the options given

1.(a + 2m)(am)
(a) a2am − 2m2
(b) a2 + am − 2m2
(c) a2 + 2m2am
(d) a2 + 2am + 2m2
2.(3ba)(2a + 3b)
(a) 6b2 + a2 − 3ab
(b) 9b2 + 3ab − 2a2
(c) 9b2 + 9ab − 3b2
(d) 6b2 + 3aba2
3.(2x + 1)2 − (x + 3)2
(a) x2 − 8
(b) x2 − 2x − 8
(c) 3x2 − 8
(d) 3x2 − 2x − 8
4.3(x + 2)2−(x − 2)2
(a) 2x2 + 16x + 8
(b) 2x2 + 16
(c) 4x2 + 8x + 16
(d) 4x2 − 16