Simplify fully:
x2 + 8x + 16x2 - 16
x2 + 8x + 16x2 - 16
Worked Solution & Answer
(x + 4)(x + 4)(x - 4)(x + 4) = x + 4x - 4 B1
Answer: x + 4x - 4
Algebraic fractions — Notes & Worked Examples Show / Hide
Explanation:
Never cancel individual terms across a + or − sign; always factorise the numerator and denominator fully first, then cancel common bracketed factors. When multiplying fractions, factorise and cancel diagonally or vertically; when dividing, flip the second fraction and multiply.
To add or subtract algebraic fractions, rewrite both fractions over their lowest common denominator (usually the product of their linear denominators), expand the numerators carefully (watching out for a minus sign in front of a bracket), and simplify.
Example 1: Simplify fully 2x2 + 5x − 3x2 − 9
Solution:
Factorise numerator: 2x2 + 6x − x − 3 = (2x − 1)(x + 3)
Factorise denominator (DOTS): x2 − 9 = (x − 3)(x + 3)
= (2x − 1)(x + 3)(x − 3)(x + 3)
Cancel common factor (x + 3):
Answer: 2x − 1x − 3
Example 2: Write 4x − 2 − 3x + 1 as a single fraction.
Solution:
Common denominator is (x − 2)(x + 1):
= 4(x + 1) − 3(x − 2)(x − 2)(x + 1)
Expand top: 4x + 4 − 3x + 6 = x + 10
Answer: x + 10(x − 2)(x + 1)
Simplify fully:
x2 + 10x + 25x2 - 25
x2 + 10x + 25x2 - 25
Worked Solution & Answer
(x + 5)(x + 5)(x - 5)(x + 5) = x + 5x - 5 B1
Answer: x + 5x - 5
Write 72a2 - 35a as a single fraction in its simplest form.
Answer
Worked Solution & Answer
Common denominator is 10a²: M1
(7 × 5) / (10a²) – (3 × 2a) / (10a²) = (35 – 6a) / (10a²). A1
(7 × 5) / (10a²) – (3 × 2a) / (10a²) = (35 – 6a) / (10a²). A1
Answer: Answer 35 - 6a10a2
Express 54 + x - 36x as a single fraction in its simplest form.
Worked Solution & Answer
15x / 12x + 2(x – 3) / 12x = (15x + 2x – 6) / 12x = (17x – 6) / 12x M2 A1
Answer: 17x - 612x
Express (20x2 - 36 - 2x - 6) × 14 - x as a single fraction in its simplest form.
Worked Solution & Answer
(20 – 2(x + 6)) / ((x – 6)(x + 6)) × 1/(4 – x) = (8 – 2x) / (x² – 36) × 1/(4 – x) = 2(4 – x) / ((x² – 36)(4 – x)) = 2 / (x² – 36) M2 A1
Answer: 2x2 - 36
Given that x = 59y + 5 and that y = 55a - 2
find an expression for x in terms of a.
Give your expression as a single fraction in its simplest form.
find an expression for x in terms of a.
Give your expression as a single fraction in its simplest form.
x =
Worked Solution & Answer
x = 5 / [9(5/(5a − 2)) + 5] = 5(5a − 2) / [45 + 5(5a − 2)] = 5(5a − 2) / (25a + 35) = 5(5a − 2) / [5(5a + 7)] = (5a − 2)/(5a + 7) M3 A1
Answer: x = x = 5a - 25a + 7
Write 4 - [(3x - 5) ÷ 3x2 + x - 104x - 1] as a single fraction in its simplest form.
Worked Solution & Answer
Factorise quadratic: 3x² + x − 10 = (3x − 5)(x + 2) M2
(3x − 5) ÷ [(3x − 5)(x + 2) / (4x − 1)] = (3x − 5) × (4x − 1) / [(3x − 5)(x + 2)] = (4x − 1) / (x + 2) M1
4 − (4x − 1) / (x + 2) = [4(x + 2) − (4x − 1)] / (x + 2) = (4x + 8 − 4x + 1) / (x + 2) = 9 / (x + 2) A1
(3x − 5) ÷ [(3x − 5)(x + 2) / (4x − 1)] = (3x − 5) × (4x − 1) / [(3x − 5)(x + 2)] = (4x − 1) / (x + 2) M1
4 − (4x − 1) / (x + 2) = [4(x + 2) − (4x − 1)] / (x + 2) = (4x + 8 − 4x + 1) / (x + 2) = 9 / (x + 2) A1
Answer: 9x + 2
Write 54x2 - 25 - (2x + 3) ÷ (4x2 + 16x + 157) as a single fraction in its simplest form.
Show clear algebraic working.
Show clear algebraic working.
Worked Solution & Answer
4x² + 16x + 15 = (2x + 3)(2x + 5) and 4x² – 25 = (2x – 5)(2x + 5) M2
Second term = (2x + 3) × 7 / ((2x + 3)(2x + 5)) = 7 / (2x + 5) M1
Then 5 / ((2x – 5)(2x + 5)) – 7(2x – 5) / ((2x – 5)(2x + 5)) = (5 – 14x + 35) / (4x² – 25) = (40 – 14x) / (4x² – 25). A1
Second term = (2x + 3) × 7 / ((2x + 3)(2x + 5)) = 7 / (2x + 5) M1
Then 5 / ((2x – 5)(2x + 5)) – 7(2x – 5) / ((2x – 5)(2x + 5)) = (5 – 14x + 35) / (4x² – 25) = (40 – 14x) / (4x² – 25). A1
Answer: 40 - 14x4x2 - 25
Express
13x - 2 × 9x2 - 43x2 - 13x - 10 - 7x - 1
as a single fraction in its simplest form.
13x - 2 × 9x2 - 43x2 - 13x - 10 - 7x - 1
as a single fraction in its simplest form.
Worked Solution & Answer
(9x² − 4) / (3x² − 13x − 10) = (3x − 2)(3x + 2) / ((3x + 2)(x − 5)) M2
Multiplying by 1/(3x − 2) gives 1/(x − 5) M2
Then 1/(x − 5) − 7/(x − 1) = (x − 1 − 7(x − 5)) / ((x − 5)(x − 1)) = (34 − 6x) / ((x − 5)(x − 1)) = 2(17 − 3x) / ((x − 5)(x − 1)) A1
Multiplying by 1/(3x − 2) gives 1/(x − 5) M2
Then 1/(x − 5) − 7/(x − 1) = (x − 1 − 7(x − 5)) / ((x − 5)(x − 1)) = (34 − 6x) / ((x − 5)(x − 1)) = 2(17 − 3x) / ((x − 5)(x − 1)) A1
Answer: 2(17 - 3x)(x - 5)(x - 1)
(a) Simplify [(4x² – 9)/(6x + 9)] × [2x/(x² – 3x)] in the form (ax + b)/(cx + d). (3) (b) Express 3/(x + 1) + 1/(x – 2) – 4/x as a single fraction. (3)
(a) Write 4x2 - 96x + 9 × 2xx2 - 3x in the form ax + bcx + d where a, b, c and d are integers. [3 marks]
(b) Express 3x + 1 + 1x - 2 - 4x as a single fraction in its simplest form. [3 marks]
Worked Solution & Answer
(a)
[(2x – 3)(2x + 3) / 3(2x + 3)] × [2x / x(x – 3)] = 2(2x – 3) / 3(x – 3) = (4x – 6)/(3x – 9) M2 A1
Answer (a): 4x - 63x - 9
(b)
[3x(x – 2) + x(x + 1) – 4(x + 1)(x – 2)] / [x(x + 1)(x – 2)] = (3x² – 6x + x² + x – 4x² + 4x + 8) / [x(x + 1)(x – 2)] = (8 – x) / [x(x + 1)(x – 2)] M2 A1
Answer (b): 8 - xx(x + 1)(x - 2)