Chapter 4: Factorization and Algebraic Manipulation

Mathematics

Class 92025–26 SNC30 Questions SolvedEnglish Medium

Exercise 4.1 — Class 9 Mathematics Solutions

All 30 questions below are worked step by step — every step clearly worked out with mathematical reasons — for the Punjab Board 9th class Mathematics textbook.

Exercise 4.1

4 Questions (26 Sub-parts)

Quick Solution Step by Step

Just the working. Switch to Step by Step for the full explanation.

Question 1

EasyApproved by Miss Aniya • Sep 8, 2026

Factorize by identifying common factors.

Solution

This question consists of 6 sub-parts. Follow the step-by-step factorization for each sub-part below by finding the highest common monomial factor (HCF).

Step by Step

Q1Part (i)

EasyApproved by Miss Aniya • Sep 8, 2026

$6 x + 12$

Quick Solution

Under Review

\begin{matrix} 6 x + 12 & = 6 (x) + 6 (2) \\ = 6 (x + 2) \end{matrix}

Step by Step

Q1Part (ii)

EasyApproved by Miss Aniya • Sep 8, 2026

$15 y^{2} + 20 y$

Quick Solution

Under Review

\begin{matrix} 15 y^{2} + 20 y & = 5 y (3 y) + 5 y (4) \\ = 5 y (3 y + 4) \end{matrix}

Step by Step

Q1Part (iii)

EasyApproved by Miss Aniya • Sep 8, 2026

$- 12 x^{2} - 3 x$

Quick Solution

Under Review

\begin{matrix} - 12 x^{2} - 3 x & = - 3 x (4 x) + (- 3 x) (1) \\ = - 3 x (4 x + 1) \end{matrix}

Step by Step

Q1Part (iv)

EasyApproved by Miss Aniya • Sep 8, 2026

$4 a^{2} b + 8 a b^{2}$

Quick Solution

Under Review

\begin{matrix} 4 a^{2} b + 8 a b^{2} & = 4 a b (a) + 4 a b (2 b) \\ = 4 a b (a + 2 b) \end{matrix}

Step by Step

Q1Part (v)

EasyApproved by Miss Aniya • Sep 8, 2026

$x y - 3 x^{2} + 2 x$

Quick Solution

Under Review

\begin{matrix} x y - 3 x^{2} + 2 x & = x (y) - x (3 x) + x (2) \\ = x (y - 3 x + 2) \end{matrix}

Step by Step

Q1Part (vi)

EasyApproved by Miss Aniya • Sep 8, 2026

$3 a^{2} b - 9 a b^{2} + 15 a b$

Quick Solution

Under Review

\begin{matrix} 3 a^{2} b - 9 a b^{2} + 15 a b & = 3 a b (a) - 3 a b (3 b) + 3 a b (5) \\ = 3 a b (a - 3 b + 5) \end{matrix}

Step by Step

Question 2

EasyUnder Review

Factorize and represent pictorially:

Solution

This question consists of 4 sub-parts. Follow the step-by-step algebraic factorization and algebra tile area model pictorial representation for each sub-part below.

Step by Step

Q2Part (i)

EasyApproved by Miss Aniya • Sep 8, 2026

$5 x + 15$

Quick Solution

Under Review

\begin{matrix} 5 x + 15 & = 5 (x) + 5 (3) \\ = 5 (x + 3) \end{matrix}

Step by Step

Q2Part (ii)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} + 4 x + 3$

Quick Solution

Under Review

\begin{matrix} x^{2} + 4 x + 3 & = x^{2} + 3 x + x + 3 \\ = x (x + 3) + 1 (x + 3) \\ = (x + 3) (x + 1) \end{matrix}

Step by Step

Q2Part (iii)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} + 6 x + 8$

Quick Solution

Under Review

\begin{matrix} x^{2} + 6 x + 8 & = x^{2} + 4 x + 2 x + 8 \\ = x (x + 4) + 2 (x + 4) \\ = (x + 4) (x + 2) \end{matrix}

Step by Step

Q2Part (iv)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} + 4 x + 4$

Quick Solution

Under Review

\begin{matrix} x^{2} + 4 x + 4 & = x^{2} + 2 x + 2 x + 4 \\ = x (x + 2) + 2 (x + 2) \\ = (x + 2) (x + 2) \\ = (x + 2)^{2} \end{matrix}

Step by Step

Question 3

MediumApproved by Miss Aniya • Sep 8, 2026

Factorize:

Solution

This question consists of 8 sub-parts. Follow the step-by-step quadratic factorization by splitting the middle term for each sub-part below.

Step by Step

Q3Part (i)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} + x - 12$

Quick Solution

Under Review

\begin{matrix} x^{2} + x - 12 & = x^{2} + 4 x - 3 x - 12 \\ = x (x + 4) - 3 (x + 4) \\ = (x + 4) (x - 3) \end{matrix}

Step by Step

Q3Part (ii)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} + 7 x + 10$

Quick Solution

Under Review

\begin{matrix} x^{2} + 7 x + 10 & = x^{2} + 5 x + 2 x + 10 \\ = x (x + 5) + 2 (x + 5) \\ = (x + 5) (x + 2) \end{matrix}

Step by Step

Q3Part (iii)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} - 6 x + 8$

Quick Solution

Under Review

\begin{matrix} x^{2} - 6 x + 8 & = x^{2} - 4 x - 2 x + 8 \\ = x (x - 4) - 2 (x - 4) \\ = (x - 4) (x - 2) \end{matrix}

Step by Step

Q3Part (iv)

MediumApproved by Miss Aniya • Sep 8, 2026

$x^{2} - x - 56$

Quick Solution

Under Review

\begin{matrix} x^{2} - x - 56 & = x^{2} - 8 x + 7 x - 56 \\ = x (x - 8) + 7 (x - 8) \\ = (x - 8) (x + 7) \end{matrix}

Step by Step

Q3Part (v)

MediumApproved by Miss Aniya • Sep 8, 2026

$x^{2} - 10 x - 24$

Quick Solution

Under Review

\begin{matrix} x^{2} - 10 x - 24 & = x^{2} - 12 x + 2 x - 24 \\ = x (x - 12) + 2 (x - 12) \\ = (x - 12) (x + 2) \end{matrix}

Step by Step

Q3Part (vi)

EasyApproved by Miss Aniya • Sep 8, 2026

$y^{2} + 4 y - 12$

Quick Solution

Under Review

\begin{matrix} y^{2} + 4 y - 12 & = y^{2} + 6 y - 2 y - 12 \\ = y (y + 6) - 2 (y + 6) \\ = (y + 6) (y - 2) \end{matrix}

Step by Step

Q3Part (vii)

MediumApproved by Miss Aniya • Sep 8, 2026

$y^{2} + 13 y + 36$

Quick Solution

Under Review

\begin{matrix} y^{2} + 13 y + 36 & = y^{2} + 9 y + 4 y + 36 \\ = y (y + 9) + 4 (y + 9) \\ = (y + 9) (y + 4) \end{matrix}

Step by Step

Q3Part (viii)

EasyApproved by Miss Aniya • Sep 8, 2026

$x^{2} - x - 2$

Quick Solution

Under Review

\begin{matrix} x^{2} - x - 2 & = x^{2} - 2 x + x - 2 \\ = x (x - 2) + 1 (x - 2) \\ = (x - 2) (x + 1) \end{matrix}

Step by Step

Question 4

MediumApproved by Miss Aniya • Sep 8, 2026

Factorize:

Solution

This question consists of 8 sub-parts. Follow the step-by-step factorization of quadratic trinomials of the form $a x^{2} + b x + c$ with $a > 1$ for each sub-part below.

Step by Step

Q4Part (i)

EasyApproved by Miss Aniya • Sep 8, 2026

$2 x^{2} + 7 x + 3$

Quick Solution

Under Review

\begin{matrix} 2 x^{2} + 7 x + 3 & = 2 x^{2} + 6 x + x + 3 \\ = 2 x (x + 3) + 1 (x + 3) \\ = (x + 3) (2 x + 1) \end{matrix}

Step by Step

Q4Part (ii)

MediumApproved by Miss Aniya • Sep 8, 2026

$2 x^{2} + 11 x + 15$

Quick Solution

Under Review

\begin{matrix} 2 x^{2} + 11 x + 15 & = 2 x^{2} + 6 x + 5 x + 15 \\ = 2 x (x + 3) + 5 (x + 3) \\ = (x + 3) (2 x + 5) \end{matrix}

Step by Step

Q4Part (iii)

MediumApproved by Miss Aniya • Sep 8, 2026

$4 x^{2} + 13 x + 3$

Quick Solution

Under Review

\begin{matrix} 4 x^{2} + 13 x + 3 & = 4 x^{2} + 12 x + x + 3 \\ = 4 x (x + 3) + 1 (x + 3) \\ = (x + 3) (4 x + 1) \end{matrix}

Step by Step

Q4Part (iv)

EasyApproved by Miss Aniya • Sep 8, 2026

$3 x^{2} + 5 x + 2$

Quick Solution

Under Review

\begin{matrix} 3 x^{2} + 5 x + 2 & = 3 x^{2} + 3 x + 2 x + 2 \\ = 3 x (x + 1) + 2 (x + 1) \\ = (x + 1) (3 x + 2) \end{matrix}

Step by Step

Q4Part (v)

MediumApproved by Miss Aniya • Sep 8, 2026

$3 y^{2} - 11 y + 6$

Quick Solution

Under Review

\begin{matrix} 3 y^{2} - 11 y + 6 & = 3 y^{2} - 9 y - 2 y + 6 \\ = 3 y (y - 3) - 2 (y - 3) \\ = (y - 3) (3 y - 2) \end{matrix}

Step by Step

Q4Part (vi)

EasyApproved by Miss Aniya • Sep 8, 2026

$2 y^{2} - 5 y + 2$

Quick Solution

Under Review

\begin{matrix} 2 y^{2} - 5 y + 2 & = 2 y^{2} - 4 y - y + 2 \\ = 2 y (y - 2) - 1 (y - 2) \\ = (y - 2) (2 y - 1) \end{matrix}

Step by Step

Q4Part (vii)

MediumApproved by Miss Aniya • Sep 8, 2026

$4 z^{2} - 11 z + 6$

Quick Solution

Under Review

\begin{matrix} 4 z^{2} - 11 z + 6 & = 4 z^{2} - 8 z - 3 z + 6 \\ = 4 z (z - 2) - 3 (z - 2) \\ = (z - 2) (4 z - 3) \end{matrix}

Step by Step

Q4Part (viii)

MediumApproved by Miss Aniya • Sep 8, 2026

$6 + 7 x - 3 x^{2}$

Quick Solution

Under Review

\begin{matrix} 6 + 7 x - 3 x^{2} & = 6 - 2 x + 9 x - 3 x^{2} \\ = 2 (3 - x) + 3 x (3 - x) \\ = (3 - x) (2 + 3 x) \end{matrix}

Step by Step

The exercises and question numbering reproduced on these pages are from Mathematics for Class 9 (National Curriculum of Pakistan 2023), published by the Punjab Curriculum and Textbook Board (PCTB), Lahore, authored by Muhammad Akhtar Shirani, Madiha Mahmood, and Ghulam Murtaza. PCTB holds the copyright in the original textbook. PrepSure is not affiliated with, endorsed by, or sponsored by PCTB. The worked solutions, explanations, hints and method notes are PrepSure's own original work, written and reviewed by our team. If you hold rights in this material and believe anything here exceeds fair use, write to us and we will take it down.

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