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Expanding Brackets and Simplifying Expressions: A Level Mathematics, Summaries of Algebra

University of DundeeAlgebra

Expanding brackets and simplifying expressions. A LEVEL LINKS. Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds.

Typology: Summaries

2021/2022

Uploaded on 09/27/2022

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Expanding brackets
and simplifying expressions
A LEVEL LINKS
Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds
Key points
When you expand one set of brackets you must multiply everything inside the bracket by
what is outside.
When you expand two linear expressions, each with two terms of the form ax + b, where
a ≠ 0 and b ≠ 0, you create four terms. Two of these can usually be simplified by collecting
like terms.
Examples
Example 1 Expand 4(3x − 2)
4(3x − 2) = 12x − 8
Multiply everything inside the bracket
by the 4 outside the bracket
Example 2 Expand and simplify 3(x + 5) − 4(2x + 3)
3(x + 5) − 4(2x + 3)
= 3x + 15 − 8x – 12
= 3 − 5x
1 Expand each set of brackets
separately by multiplying (x + 5) by
3 and (2x + 3) by −4
2 Simplify by collecting like terms:
3x − 8x = −5x and 15 − 12 = 3
Example 3 Expand and simplify (x + 3)(x + 2)
(x + 3)(x + 2)
= x(x + 2) + 3(x + 2)
= x2 + 2x + 3x + 6
= x2 + 5x + 6
1 Expand the brackets by multiplying
(x + 2) by x and (x + 2) by 3
2 Simplify by collecting like terms:
2x + 3x = 5x
Example 4 Expand and simplify (x − 5)(2x + 3)
(x − 5)(2x + 3)
= x(2x + 3) − 5(2x + 3)
= 2x2 + 3x − 10x − 15
= 2x2 − 7x − 15
1 Expand the brackets by multiplying
(2x + 3) by x and (2x + 3) by −5
2 Simplify by collecting like terms:
3x − 10x = −7x
pf3

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Expanding brackets

and simplifying expressions

A LEVEL LINKS

Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds

Key points

 When you expand one set of brackets you must multiply everything inside the bracket by what is outside.  When you expand two linear expressions, each with two terms of the form ax + b , where a ≠ 0 and b ≠ 0, you create four terms. Two of these can usually be simplified by collecting like terms.

Examples

Example 1 Expand 4(3 x − 2)

4(3 x − 2) = 12 x − 8 Multiply everything inside the bracket by the 4 outside the bracket

Example 2 Expand and simplify 3( x + 5) − 4(2 x + 3)

3( x + 5) − 4(2 x + 3) = 3 x + 15 − 8 x – 12

= 3 − 5 x

1 Expand each set of brackets separately by multiplying ( x + 5) by 3 and (2 x + 3) by − 4

2 Simplify by collecting like terms: 3 x − 8 x = − 5 x and 15 − 12 = 3

Example 3 Expand and simplify ( x + 3)( x + 2)

( x + 3)( x + 2) = x ( x + 2) + 3( x + 2) = x^2 + 2 x + 3 x + 6 = x^2 + 5 x + 6

1 Expand the brackets by multiplying ( x + 2) by x and ( x + 2) by 3

2 Simplify by collecting like terms: 2 x + 3 x = 5 x

Example 4 Expand and simplify ( x − 5)(2 x + 3)

( x − 5)(2 x + 3) = x (2 x + 3) − 5(2 x + 3) = 2 x^2 + 3 x − 10 x − 15 = 2 x^2 − 7 x − 15

1 Expand the brackets by multiplying (2 x + 3) by x and (2 x + 3) by − 5

2 Simplify by collecting like terms: 3 x − 10 x = − 7 x

Practice

1 Expand.

a 3(2 x − 1) b −2(5 pq + 4 q^2 ) c −(3 xy − 2 y^2 )

2 Expand and simplify.

a 7(3 x + 5) + 6(2 x – 8) b 8(5 p – 2) – 3(4 p + 9) c 9(3 s + 1) – 5(6 s – 10) d 2(4 x – 3) – (3 x + 5)

3 Expand.

a 3 x (4 x + 8) b 4 k (5 k^2 – 12) c – 2 h (6 h^2 + 11 h – 5) d – 3 s (4 s^2 – 7 s + 2)

4 Expand and simplify.

a 3( y^2 – 8) – 4( y^2 – 5) b 2 x ( x + 5) + 3 x ( x – 7) c 4 p (2 p – 1) – 3 p (5 p – 2) d 3 b (4 b – 3) – b (6 b – 9)

5 Expand 12 (2 y – 8)

6 Expand and simplify.

a 13 – 2( m + 7) b 5 p ( p^2 + 6 p ) – 9 p (2 p – 3)

7 The diagram shows a rectangle.

Write down an expression, in terms of x , for the area of the rectangle. Show that the area of the rectangle can be written as 21 x^2 – 35 x

8 Expand and simplify.

a ( x + 4)( x + 5) b ( x + 7)( x + 3) c ( x + 7)( x – 2) d ( x + 5)( x – 5) e (2 x + 3)( x – 1) f (3 x – 2)(2 x + 1) g (5 x – 3)(2 x – 5) h (3 x – 2)(7 + 4 x ) i (3 x + 4 y )(5 y + 6 x ) j ( x + 5)^2 k (2 x − 7)^2 l (4 x − 3 y )^2

Extend

9 Expand and simplify ( x + 3)² + ( x − 4)²

10 Expand and simplify.

a

x x

x x

b

2

x

x

Watch out!

When multiplying (or dividing) positive and negative numbers, if the signs are the same the answer is ‘+’; if the signs are different the answer is ‘–’.