Algebraic Fractions (Copy)

Practice Questions — 2.3 Algebraic Fractions

Question 1

Simplify:
(a) (x³ × x²) ÷ x⁴
(b) (3a⁴ × 9a¹⁰)
(c) (3a⁴ ÷ 9a¹⁰)

Question 2

Simplify:
(a) (y² − 9) ÷ (y − 3)
(b) (x² − 16) ÷ (x − 4)
(c) (p² − 25) ÷ (p + 5)

Question 3

Simplify fully:
(a) (x² − 2x) ÷ (x² − 5x)
(b) (3a² − 6a) ÷ (a² − 2a)
(c) (y² + 4y + 4) ÷ (y + 2)

Question 4

Simplify:
(a) (x² − 9) ÷ (x² − x − 6)
(b) (m² − 9m + 18) ÷ (m² − 6m + 9)
(c) (t² − 36) ÷ (t² + 12t + 36)

Question 5

Simplify:
(a) (x² − 6x + 9) ÷ (x − 3)
(b) (n² − 4n) ÷ (n² − 2n)
(c) (p² − q²) ÷ (p − q)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

Question 6

Simplify and express with a single denominator:
(a) 1/(x − 2) + (x + 1)/(x − 3)
(b) 2/(y + 1) − 3/(y − 1)
(c) (p + 2)/(p − 1) + (2p − 3)/(p − 1)

Question 7

Simplify:
(a) (x² − 25)/(x² − 5x)
(b) (y² − 49)/(y² + 7y)
(c) (a² − 1)/(a² + a)

Question 8

Simplify fully:
(a) (2x² − 8x)/(4x)
(b) (3y² − 6y)/(9y)
(c) (5p² + 10p)/(15p)

Question 9

Simplify:
(a) (2x² + 8x)/(x² + 4x)
(b) (m² + 2m)/(m² − 4)
(c) (k² − 9k)/(k² − 3k)

Question 10

Simplify and leave in lowest terms:
(a) (x³ − 27)/(x² − 3x)
(b) (y³ − 8)/(y² − 2y)
(c) (p³ − q³)/(p² − pq)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

Answer key and explanations — 2.3 Algebraic Fractions

1.

(a) (x³ × x²) ÷ x⁴ = x⁵ ÷ x⁴ = x
(b) (3a⁴ × 9a¹⁰) = 27a¹⁴
(c) (3a⁴ ÷ 9a¹⁰) = (3/9) × a⁴⁻¹⁰ = (1/3)a⁻⁶ = 1/(3a⁶)
Explanation: Apply laws of indices (aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ).

2.

(a) (y² − 9)/(y − 3) = (y − 3)(y + 3)/(y − 3) = y + 3
(b) (x² − 16)/(x − 4) = (x − 4)(x + 4)/(x − 4) = x + 4
(c) (p² − 25)/(p + 5) = (p − 5)(p + 5)/(p + 5) = p − 5
Explanation: Factorise numerator, cancel common terms.

3.

(a) (x² − 2x)/(x² − 5x) = x(x − 2)/x(x − 5) = (x − 2)/(x − 5)
(b) (3a² − 6a)/(a² − 2a) = 3a(a − 2)/a(a − 2) = 3
(c) (y² + 4y + 4)/(y + 2) = (y + 2)(y + 2)/(y + 2) = y + 2
Explanation: Factorise and cancel.

4.

(a) (x² − 9)/(x² − x − 6) = (x − 3)(x + 3)/[(x − 3)(x + 2)] = (x + 3)/(x + 2)
(b) (m² − 9m + 18)/(m² − 6m + 9) = (m − 3)(m − 6)/(m − 3)(m − 3) = (m − 6)/(m − 3)
(c) (t² − 36)/(t² + 12t + 36) = (t − 6)(t + 6)/(t + 6)(t + 6) = (t − 6)/(t + 6)
Explanation: Recognise difference of squares and quadratics.

5.

(a) (x² − 6x + 9)/(x − 3) = (x − 3)(x − 3)/(x − 3) = x − 3
(b) (n² − 4n)/(n² − 2n) = n(n − 4)/n(n − 2) = (n − 4)/(n − 2)
(c) (p² − q²)/(p − q) = (p − q)(p + q)/(p − q) = p + q

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

6.

(a) 1/(x − 2) + (x + 1)/(x − 3)
= [(x − 3) + (x + 1)(x − 2)] / [(x − 2)(x − 3)]
= (x − 3 + x² − x − 2)/(x² − 5x + 6)
= (x² − 5)/(x² − 5x + 6)

(b) 2/(y + 1) − 3/(y − 1)
= [2(y − 1) − 3(y + 1)]/[(y + 1)(y − 1)]
= (2y − 2 − 3y − 3)/(y² − 1) = (−y − 5)/(y² − 1)

(c) (p + 2)/(p − 1) + (2p − 3)/(p − 1)
= [(p + 2) + (2p − 3)]/(p − 1) = (3p − 1)/(p − 1)

7.

(a) (x² − 25)/(x² − 5x) = (x − 5)(x + 5)/x(x − 5) = (x + 5)/x
(b) (y² − 49)/(y² + 7y) = (y − 7)(y + 7)/y(y + 7) = (y − 7)/y
(c) (a² − 1)/(a² + a) = (a − 1)(a + 1)/a(a + 1) = (a − 1)/a

8.

(a) (2x² − 8x)/(4x) = 2x(x − 4)/(4x) = (x − 4)/2
(b) (3y² − 6y)/(9y) = 3y(y − 2)/(9y) = (y − 2)/3
(c) (5p² + 10p)/(15p) = 5p(p + 2)/(15p) = (p + 2)/3

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

9.

(a) (2x² + 8x)/(x² + 4x) = 2x(x + 4)/x(x + 4) = 2
(b) (m² + 2m)/(m² − 4) = m(m + 2)/(m − 2)(m + 2) = m/(m − 2)
(c) (k² − 9k)/(k² − 3k) = k(k − 9)/k(k − 3) = (k − 9)/(k − 3)

10.

(a) (x³ − 27)/(x² − 3x) = (x − 3)(x² + 3x + 9)/x(x − 3) = (x² + 3x + 9)/x
(b) (y³ − 8)/(y² − 2y) = (y − 2)(y² + 2y + 4)/y(y − 2) = (y² + 2y + 4)/y
(c) (p³ − q³)/(p² − pq) = (p − q)(p² + pq + q²)/p(p − q) = (p² + pq + q²)/p

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course