Factorization: Comprehensive Examination
Total Marks
Time Allowed
Total Questions
Date
45
0.5 Hours
18
June 15, 2026
Section A
Section B
Section C
MCQs
Short Questions
Long Question
10 × 1 = 10 marks
7 × 3 = 21 marks
1 × 14 = 14 marks
Student Name:
Roll No. / Class:
General Instructions:
1. Answer ALL questions.
2. Show complete working for Sections B and C.
3. Cutting and overwriting should be avoided.
SECTION C — Long Question [14 Marks]
Answer the following question. Show all steps clearly.
Given: A square-shaped machine surface has area (25x²  30x + 9) m² and a cubical oil tank has volume (125x³  150x² + 60x  8)
m³
(a)
Factorize 25x²  30x + 9 completely and hence find the side length of the machine surface.
[3]
(b)
Write expressions for (i) perimeter of the square surface, and (ii) cost of edging 2 sides @ Rs 28/m.
[3]
(c)
Find the cost of polishing the entire machine surface @ Rs 75/m².
[1]
(d)
Factorize 125x³ − 150x² + 60x − 8 completely and find the expression for the side of the cubical tank.
[3]
(e)
Find (i) the total surface area of the oil tank, and (ii) the cost of painting it from outside @ Rs 32/m².
[4]
Factorization — Comprehensive Examination
(d)
(e)
(f)
Factorize
125x³
 150x²
+ of
60x
 oil
8 completely
find
the
sidethe
of
the
cubical
tank.
Find
If
x = (i)
2 m,
total
calculate
surface
the
area
actual
theside
tank
of both
andstructures
(ii)and
cost
ofand
painting
find
from
ratio
outside
of their
@
sides.
Rs 32/m².
Page 1
[2]
[1]
SECTION A — Multiple Choice Questions [10 Marks]
Circle the best answer. Each question carries 1 mark.
Which identity is used to factorize 𝑎³ − 𝑏³?
1
The factorization of 𝑥³ − 125 is:
2
8𝑥³ + 1 factorizes as:
3
64(𝑥 + 𝑦)3 − 𝑧³ factorizes as:
4
𝑥² − 21𝑥 + 90 factorizes as:
5
−3𝑦 2 + 13𝑦 − 4 factorizes as:
6
8 + 6𝑥 − 5𝑥² factorizes as:
7
8
2𝑎² − 4𝑎 − 6 fully factorized (including common
factor) is:
9
𝑝³ − 9𝑝2 𝑞 + 27𝑝𝑞2 − 27𝑞³ is the expansion of:
10
For LCM of
𝑥² + 𝑥 − 20, 𝑥 2 − 10𝑥 + 24, 𝑥 2 − 𝑥 − 30
which factor is common to ALL three?
Factorization — Comprehensive Examination
(𝑎 − 𝑏)(𝑎 + 𝑏)
(𝑎 − 𝑏)(𝑎² + 𝑎𝑏 + 𝑏²)
(𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏²)
(𝑎 + 𝑏)²
(𝑥 + 5)(𝑥² + 5𝑥 + 25)
(𝑥 + 5)(𝑥 2 − 5𝑥 + 25)
(𝑥 − 5)(𝑥 2 + 5𝑥 + 25)
(𝑥 − 5)³
(2𝑥 − 1)(4𝑥² + 2𝑥 + 1)
(2𝑥 + 1)(4𝑥 2 − 2𝑥 + 1)
(2𝑥 + 1)(4𝑥² + 2𝑥 + 1)
(2𝑥 + 1)³
(4(𝑥 + 𝑦) − 𝑧)(16(𝑥 + 𝑦)² + 4(𝑥 + 𝑦)𝑧 + 𝑧²)
(4(𝑥 + 𝑦) + 𝑧)(16(𝑥 + 𝑦 − 𝑧)
(4(𝑥 + 𝑦) − 𝑧)³
(4𝑥 + 4𝑦 − 𝑧)(16𝑥² + 𝑧²)
(𝑥 − 15)(𝑥 − 6)
(𝑥 − 18)(𝑥 − 3)
(𝑥 − 9)(𝑥 + 10)
(𝑥 − 12)(𝑥 − 9)
−(3𝑦 + 1)(𝑦 − 4)
−(3𝑦 + 1)(𝑦 − 4)
(3𝑦 − 1)(4 + 𝑦)
(3𝑦 + 1)(𝑦 − 4)
(2 + 𝑥)(4 + 5𝑥)
(2 + 𝑥)(4 − 5𝑥)
(4 + 5𝑥)(2 − 𝑥)
(4 − 5𝑥)(−2 − 𝑥)
2(𝑎 − 3)(𝑎 + 1)
(2𝑎 + 2)(𝑎 − 3)
2(𝑎 − 3)(𝑎 + 1)
(𝑎 − 3)(2𝑎 + 2)
(𝑝 − 3𝑞)³
(𝑝 + 3𝑞)³
𝑝³ − (3𝑞)³
(3𝑞 − 𝑝)³
(𝑥 − 4)
(𝑥 + 5)
(𝑥 − 6)
(𝑥 − 5)
Page 2
SECTION B — Short Questions [21 Marks]
1
Show that 𝑎6 + 729 can be written as (a) sum of two squares and (b) sum of two cubes. Write both forms.
[3]
2
Factorize completely: 3𝑝³𝑞³ − 81𝑥³
[3]
Hint: Take out common factor first, then apply difference of cubes
3
Factorize by mid-term break: 10z²  29z + 10
[3]
4
Find HCF of 12x² + x  1 and 15x² + 8x + 1 by factorization
[3]
5
Express 8 + 12t + 6t² + t³ as a product of three factors. Is each factor binomial or trinomial?
[3]
6
Factorize 27 + 512x³ and identify the values of a and b used in the formula a³ + b³
[3]
7
Factorize u4  13u² + 36 completely into linear factors
[3]
Factorization — Comprehensive Examination
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Factorization — Comprehensive Examination
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Factorization — Comprehensive Examination
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Factorization — Comprehensive Examination
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