Unit 2 | 8 Algebraic fractions and algebraic proof | Student Book chapter 11
21
Unit 2 | 8 Algebraic fractions and algebraic
proof
Key Points
To simplify algebraic fractions, first factorise the numerator and denominator if
possible and then divide the numerator and denominator by any common factors.
To add or subtract algebraic fractions, find a common denominator and then add or
subtract the numerators, just as you would for numerical fractions.
To multiply algebraic fractions, multiply the numerators and multiply the
denominators.
To divide algebraic fractions, multiply the first fraction by the reciprocal of the second
fraction (turn the second fraction upside down).
Algebraic expressions should always be given in their simplest form.
Algebra can be used to prove statements.
1
2
3
4
Simplify fully:
2(x 2 2)
a __________
x2 2 5x 1 6
3x 1 6
b __________
x2 2 2x 2 8
x2 1 4x 2 5
c ___________
x2 1 7x 1 10
4x2 2 12x
d ___________
2
x 2 9x 1 18
Simplify fully:
2 1 _____
1
a __
x x13
x
3 3 _____
b _____
x11 x22
x25
6  _____
c __
2x
x
1
4 2 _____
d _____
x15 x22
2x 2 6  _____
x23
e ______
x14
x
1
4 1 _____
f ______
2x 2 1 x 1 1
3 2 ______
2
g _____
x 2 1 2x 1 4
5x 1 25
3 3 _______
h _____
x 1 5 12x 1 6
Simplify fully:
x2 2 9
a ___________
x2 2 7x 2 30
2y2 2 10y 2 12
b _____________
3y3 2 3y
12p2 1 11p 2 5
c _____________
4p2 2 11p 2 20
4a2 2 b2
d ______________
2a2 2 5ab 2 3b2
Simplify fully:
5 1 ________
3
a ___
2x 2x2 1 8x
x2 1 3x 2 10
x2 2 3x 2 4 3 ___________
b __________
x2 1 6x 1 5
x2 2 4
1
2
c _______
2 _______
6x 2 15 4x2 2 25
2x2 1 x 2 3  ___________
x2 2 1
d __________
5x2 2 3x
5x2 1 2x 2 3
2 2 ______
12
e 3 2 _____
x 1 3 x2 2 9
3
1
f ___________
1 ___________
x2 1 4x 2 21 x2 2 7x 1 12
You learnt to factorise
quadratic expressions
in Section 8.4.
Questions in
this chapter are
targeted at the
grades indicated.
A
22
A
Unit 2 | 8 Algebraic fractions and algebraic proof | Student Book chapter 11
AO3
AO3
AO3
AO3
AO3
AO3
AO3
5
Prove that the sum of the squares of any two consecutive integers is always an odd
number.
6
Karen says she can prove that the sum of an odd number and an even number is always
odd. She says that 2 1 3 5 5 and as 5 is an odd number this proves that an odd number
added to an even number is an odd number. This isn’t a proof. Explain why.
7
Prove that (5n 1 1)2 2 (5n 2 1)2 is a multiple of 4 for all positive integer values of n.
8
Prove that the difference between the squares of any two consecutive odd numbers is
always an even number.
9
Prove that (2n 1 3)2 2 (2n 2 3)2 is always a multiple of 6.
10
Prove that the difference between the cubes of any two consecutive odd numbers is
always an even number.
11
Prove that (an 1 b)2 2 (an 2 b)2, where a and b are positive integers, is a multiple of 4
for all positive integer values of n.
Challenge yourself
This is harder than anything you will encounter in the exam, but the underlying maths is covered in your
GCSE course. Have a go and see how you do.
1
A farmer has 600 m of fencing. He is going to use the fencing to make a rectangular
sheep pen. One side of the sheep pen will be along an existing straight hedge.
Let x represent the length of the shorter side of the rectangle.
x
x
a Write down an expression, in terms of x for the longer side of the rectangle.
b Write down an equation for the area, A, of the sheep pen.
2
A ball is thrown in the air. After t seconds its height, h metres above the ground, is given by
the equation
h 5 10t 2 1 2 5t2
Find the maximum height above the ground reached by the ball.