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This pdf includes the following topics:-

Expanding brackets

Simplifying expression

Key points

Examples

Practice Problems

Expanding brackets

Simplifying expression

Key points

Examples

Practice Problems

1.
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.

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) 1 Expand each set of brackets

= 3x + 15 − 8x – 12 separately by multiplying (x + 5) by

3 and (2x + 3) by −4

= 3 − 5x 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) 1 Expand the brackets by multiplying

= x(x + 2) + 3(x + 2) (x + 2) by x and (x + 2) by 3

= x2 + 2x + 3x + 6

= x2 + 5x + 6 2 Simplify by collecting like terms:

2x + 3x = 5x

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

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

= x(2x + 3) − 5(2x + 3) (2x + 3) by x and (2x + 3) by −5

= 2x2 + 3x − 10x − 15

= 2x2 − 7x − 15 2 Simplify by collecting like terms:

3x − 10x = −7x

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.

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) 1 Expand each set of brackets

= 3x + 15 − 8x – 12 separately by multiplying (x + 5) by

3 and (2x + 3) by −4

= 3 − 5x 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) 1 Expand the brackets by multiplying

= x(x + 2) + 3(x + 2) (x + 2) by x and (x + 2) by 3

= x2 + 2x + 3x + 6

= x2 + 5x + 6 2 Simplify by collecting like terms:

2x + 3x = 5x

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

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

= x(2x + 3) − 5(2x + 3) (2x + 3) by x and (2x + 3) by −5

= 2x2 + 3x − 10x − 15

= 2x2 − 7x − 15 2 Simplify by collecting like terms:

3x − 10x = −7x

2.
1 Expand. Watch out!

a 3(2x − 1) b −2(5pq + 4q2)

When multiplying (or

c −(3xy − 2y2)

dividing) positive and

2 Expand and simplify. negative numbers, if

the signs are the same

a 7(3x + 5) + 6(2x – 8) b 8(5p – 2) – 3(4p + 9)

the answer is ‘+’; if the

c 9(3s + 1) –5(6s – 10) d 2(4x – 3) – (3x + 5) signs are different the

answer is ‘–’.

3 Expand.

a 3x(4x + 8) b 4k(5k2 – 12)

c –2h(6h2 + 11h – 5) d –3s(4s2 – 7s + 2)

4 Expand and simplify.

a 3(y2 – 8) – 4(y2 – 5) b 2x(x + 5) + 3x(x – 7)

c 4p(2p – 1) – 3p(5p – 2) d 3b(4b – 3) – b(6b – 9)

5 Expand 12 (2y – 8)

6 Expand and simplify.

a 13 – 2(m + 7) b 5p(p2 + 6p) – 9p(2p – 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

21x2 – 35x

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 (2x + 3)(x – 1) f (3x – 2)(2x + 1)

g (5x – 3)(2x – 5) h (3x – 2)(7 + 4x)

i (3x + 4y)(5y + 6x) j (x + 5)2

k (2x − 7)2 l (4x − 3y)2

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

10 Expand and simplify.

2

1 2 1

a x x b x

x x x

a 3(2x − 1) b −2(5pq + 4q2)

When multiplying (or

c −(3xy − 2y2)

dividing) positive and

2 Expand and simplify. negative numbers, if

the signs are the same

a 7(3x + 5) + 6(2x – 8) b 8(5p – 2) – 3(4p + 9)

the answer is ‘+’; if the

c 9(3s + 1) –5(6s – 10) d 2(4x – 3) – (3x + 5) signs are different the

answer is ‘–’.

3 Expand.

a 3x(4x + 8) b 4k(5k2 – 12)

c –2h(6h2 + 11h – 5) d –3s(4s2 – 7s + 2)

4 Expand and simplify.

a 3(y2 – 8) – 4(y2 – 5) b 2x(x + 5) + 3x(x – 7)

c 4p(2p – 1) – 3p(5p – 2) d 3b(4b – 3) – b(6b – 9)

5 Expand 12 (2y – 8)

6 Expand and simplify.

a 13 – 2(m + 7) b 5p(p2 + 6p) – 9p(2p – 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

21x2 – 35x

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 (2x + 3)(x – 1) f (3x – 2)(2x + 1)

g (5x – 3)(2x – 5) h (3x – 2)(7 + 4x)

i (3x + 4y)(5y + 6x) j (x + 5)2

k (2x − 7)2 l (4x − 3y)2

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

10 Expand and simplify.

2

1 2 1

a x x b x

x x x

3.
1 a 6x – 3 b –10pq – 8q2

c –3xy + 2y2

2 a 21x + 35 + 12x – 48 = 33x – 13

b 40p – 16 – 12p – 27 = 28p – 43

c 27s + 9 – 30s + 50 = –3s + 59 = 59 – 3s

d 8x – 6 – 3x – 5 = 5x – 11

3 a 12x2 + 24x b 20k3 – 48k

c 10h – 12h3 – 22h2 d 21s2 – 21s3 – 6s

4 a –y2 – 4 b 5x2 – 11x

c 2p – 7p2 d 6b2

5 y–4

6 a –1 – 2m b 5p3 + 12p2 + 27p

7 7x(3x – 5) = 21x2 – 35x

8 a x2 + 9x + 20 b x2 + 10x + 21

c x2 + 5x – 14 d x2 – 25

e 2x2 + x – 3 f 6x2 – x – 2

g 10x2 – 31x + 15 h 12x2 + 13x – 14

i 18x2 + 39xy + 20y2 j x2 + 10x + 25

k 4x2 − 28x + 49 l 16x2 − 24xy + 9y2

9 2x2 − 2x + 25

2 1

10 a x2 1 b x2 2

x2 x2

c –3xy + 2y2

2 a 21x + 35 + 12x – 48 = 33x – 13

b 40p – 16 – 12p – 27 = 28p – 43

c 27s + 9 – 30s + 50 = –3s + 59 = 59 – 3s

d 8x – 6 – 3x – 5 = 5x – 11

3 a 12x2 + 24x b 20k3 – 48k

c 10h – 12h3 – 22h2 d 21s2 – 21s3 – 6s

4 a –y2 – 4 b 5x2 – 11x

c 2p – 7p2 d 6b2

5 y–4

6 a –1 – 2m b 5p3 + 12p2 + 27p

7 7x(3x – 5) = 21x2 – 35x

8 a x2 + 9x + 20 b x2 + 10x + 21

c x2 + 5x – 14 d x2 – 25

e 2x2 + x – 3 f 6x2 – x – 2

g 10x2 – 31x + 15 h 12x2 + 13x – 14

i 18x2 + 39xy + 20y2 j x2 + 10x + 25

k 4x2 − 28x + 49 l 16x2 − 24xy + 9y2

9 2x2 − 2x + 25

2 1

10 a x2 1 b x2 2

x2 x2