Algebra and functions - Pearson Schools

Algebra and functions

1

Key points to remember 1

You can simplify expressions by collecting like terms.

2

You can simplify expressions by using the rules of indices. (i)

am
an  amn

(ii) am  an  amn 1 (iii) am   am 1

n

m

(v) a m 
a n (vi) (am)n  amn (vii) ae  1

m

(iv) a m 
a 3

You can expand an expression by multiplying each term inside the bracket by the terms outside the bracket.

4

Factorising expressions is the opposite of expanding expressions.

5

A quadratic expression has the form ax2  bx  c, where a, b and c are constants and a  0.

6

x2  y2  (x  y)(x  y). This is called a difference of squares.

7

You can write a number exactly using surds.

8

The square root of a prime number is a surd.

9

You can manipulate surds using these rules: a
a    .  
a

b (ii) (i)
ab b
b  The rules to rationalise surds are: 1 (i) for fractions in the form  , multiply the top and bottom by
a 
a



10

1 (ii) for fractions in the form  , multiply the top and bottom by a 
b  a 
b 1 (iii) for fractions in the form  , multiply the top and bottom by a 
b . a 
b

2 Algebra and functions

Example 1 Simplify (a) 2x3
3x4
5x7

(b) (2y4)3  4y2

(a) 2x3
3x4
5x7  2
3
5
x3
x4
x7  30
x347  30x14 (b) (2y4)3  4y2  (2y4)3  23
y4
3  4y2  84
y4
3  y2  2
y122  2y10

Rewrite the expression with the numbers together and the x-terms together. Using 2 (i): x3
x4
x7  x347 Using 2 (vi) 23  8 Using 2 (ii): y12  y2  y122

Example 2 Factorise (a) 6x2  x  15

(b) 4x2  9

(a) 6x  x  15 ac  90 10  9  1 So 6x2  10x  9x  15 2x(3x  5)  3(3x  5) (3x  5)(2x  3) 2

(b) 4x2  9  (2x)2  32  (2x  3)(2x  3)

This is a quadratic, where a  6, b  1 and c  15. You need to find two brackets that multiply together to give bx2  x  15. To do this work out ac Work out the two factors of ac that add to give you b Factorise using 4 Factorise by (3x  5) Using 6

Example 3 Evaluate 2 (a) 27 3 2

(b) 3

(a) 27 3  (
27 )2  (3)2 9

(8116) Using 2 (v) 3
3
3  27 So 27  3

Algebra and functions 3

(b)



  81  16

2 3

 

16   81

2 3

Using 2 (iii)

4

(
16 )3  4 (
81 )3

Using 2 (v)

23  3 3

As 2
2
2
2  16 and 3
3
3
3  81

8 = 27

Example 4 Simplify (a)
40 

(b)
40  
80 

(a) 40 
4 
10  2
10  (b)
40  
180   2
10  
9 

20   2
10   3
20   5
10 

Using 9
4 2 Using 9

Worked exam style question 1

(a) Express 80 in the form a
5, where a is an integer. (b) Express (4 
5)2 in the form b  c
5, where b and c are integers.

(a)
80  
16 
5  4
5  (b) (4 
5 )2 means (4 
5 )(4 
5 )  16  8
5   (
5 )2 4  16  4
5  
5   (5)2  16  8
5 5   21  8
5

(
5 )2  5

4 Algebra and functions

Worked exam style question 2 Rationalise the denominator for
3 
5\
5 
3 

(
3  
5 )(
5  
3 )  (
5  
3 )(
5  
3 )
3 

5  
5 
5  
3 
3  
5 
3   
5 
5  
3 
5  
5 
3  
3 
3  
15  
25  
9 
15   
25 
15 
15  
9 

Using 10 : Multiply top and bottom by (
5  +
3 ) Using 9

You should be able to miss this step out and write this straight down

5  3  53 2  2  1

Worked exam style question 3

(a) Given that 27  3k, write down the value of k. (b) Given that 9y  272y1, find the value of y.

(a) 27  3k 33  3k k3 9y  272y1 (32)y  (33)2y1 32y  36y3

(b)

So

2y  6y  3 4y  3 y  34

Comparing powers

Using 2 (vi) Comparing powers

Algebra and functions 5

Revision exercise 1 1 Simplify (a) 3x2
4x5
2x3

(b) (8y4)2  4y3

2 Simplify (a) 9x2  4y2

(b) 6y2  15y

(c) 12x2  14x  6

(d) 2x2  5x  3

(e) 6  x  2x2 3 Evaluate 1

2

(a) 16 2

(b) 64 3

(d)

(245)

1 2

(e)

(287)

2

(c) (343) 3 2 3

(e)

(287)

2 3

4 Evaluate
245   3
45   220, giving your answer in terms of a
5 where a is a constant. 5 Simplify (a)
2

8

(b)
6

8

12 

6 Rationalise the denominators of 1
2 (a)  (b)  3 
5
6 
2

3
5 2 (c)  2
5 4

1 1 7 Simplify   
3  1
3  1 8 (a) Express
112  in the form a
7, where
a is an integer. (b) Express (3  7)2 in the form b  c
5, where b and c are integers. 9 (a) Given that 8  2k, write down the value of k. (b) Given that 4x  82x, find the values of x.

E

10 Find the value of 1

3

(a) 81 2

(b) 81 4

3

(c) 81 4 E

6 Algebra and functions

Test yourself

What to review If your answer is incorrect

27  8

1 Evaluate (

)

2 3

Review Heinemann Book C1 pages 10–11 Revise for C1 page 2 Example 3

2 Simplify the expressions (a) 3(x  4y2)  2(3x  y2) (b) 4x2
3x5  6x3

Review Heinemann Book C1 page 1 Review Heinemann Book C1 pages 2–3 Revise for C1 page 2 Example 1

3 Expand 5x2(3x  2)  3x2(2x  5)

Review Heinemann Book C1 pages 3–4

4 Factorise completely (a) 4x2  10x (b) 16x2  9y2 (c) 6x2  7x  5

Review Heinemann Book C1 page 4 Review Heinemann Book C1 pages 4–6 Revise for C1 page 2 Example 2b Review Heinemann Book C1 pages 5–6 Revise for C1 page 2 Example 2a

5 Simplify (a)
172  (b) 2
12  
48   3
75 

Review Heinemann Book C1 pages 9–10 Revise for C1 page 3 Example 4

6 Rationalise the denominators of
5  2 4 (a)  (b)  1 
3
7  3

Review Heinemann Book C1 pages 10–11 Revise for C1 page 4 Worked exam style question 2

7 Express (3 
11 )2 in the form a  b
11 

Review Heinemann Book C1 pages 10–11 Revise for C1 page 3 Worked exam style question 1

Test yourself answers 1

9  4

2 (a) 10y2  3x (b) 2x4

5 (a) 6
2 (b) 23
3

3 9x3  5x2

6 (a) 2(1  3) (b)

4 (a) 2x(2x  5) (b) (4x  3y)(4x  3y) (c) (2x  1)(3x  5) 1  2

(6 
35   2
7   3
5 )

7 20 + 6
11