Class 10 Maths Chapter 7
Factorization
Important Questions

Class 10 Mathematics Chapter 7, Factorization is a crucial topic in ICSE Class 10 mathematics. It involves breaking down algebraic expressions or numbers into their constituent factors. Understanding factorization is essential as it forms the foundation for many other mathematical concepts. Here are some factorisation class 10 ICSE important questions.

Introduction

Factorization as we study in  factorisation class 10 ICSE   is the process of expressing a number or algebraic expression as the product of its factors. Factors are numbers or expressions that multiply together to yield the original number or expression. The main goal of factorization is to simplify complex expressions, solve equations, and understand the underlying structure of mathematical concepts.
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Common Factorization Techniques:

Common Factors: Identifying and factoring out the common factors shared by multiple terms in an expression.
Example: Factorize 6x + 9y. The common factor is 3, so the expression becomes 3(2x + 3y).
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Difference of Squares: Recognizing expressions in the form of a² - b² and factoring them as (a + b)(a - b).
Example: Factorize x² - 9. The expression factors as (x + 3)(x - 3).

What is Factorization?

In ICSE Class 10 mathematics, factorization refers to the process of expressing an algebraic expression or a number as the product of its constituent factors. Factors are numbers or algebraic expressions that, when multiplied together, result in the original expression or number. Factorization is an essential skill in mathematics and is used to simplify expressions, solve equations, and understand the underlying structure of mathematical concepts.

For example, consider the algebraic expression 2x² + 4x. Factorization of this expression involves finding two expressions that, when multiplied, result in 2x² + 4x. In this case, the factors are 2x and (x + 2), so the factorization is:

2x² + 4x = 2x(x + 2).
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Factorization is also used to break down numbers into their prime factors. For instance, the prime factorization of the number 12 is:
12 = 2 * 2 * 3.

Class 10 Factorization Important Questions and Answers

Q1. What is the remainder, if we divide 6x³+ x² – 2x + 4 by x – 2 ?

Options

(a) 48
(b) 52
(c) -26
(d) -24

Ans. (b) 52

Explanation:
Let
p(x) = 6x³+ x² – 2x + 4
When, p(x) is divided by x – 2,
Remainder = p(x = 2)
= 6(2)³ + (2)²– 2(2) + 4
= 48 + 4 – 4 + 4 = 52

Q2. f (x + 2) is a factor of 3x³– x² – px – 4, then the value of p is:

Options

(a) 14
(b) 12
(c) 10
(d) 16

Ans. (d) 16

Explanation:
Let f(x) = 3x³– x²  – px – 4 ...(i)
Since, (x + 2) is a factor of f(x), f(–2) = 0
⇒ 3(–2)³ – (–2)³  – p(–2) – 4 = 0
⇒ –24 – 4 + 2p – 4 = 0
⇒ 2p = 32
⇒ p = 16

Q3. If x – 2 is a factor of 2x³ – x² – px – 2.
(i) find the value of p
(ii) with the value of p, factorize the above expression completely.

Explanation:
Given expression is 2x³ – x² – px – 2 and x – 2 is the factor.
(i) x – 2 = 0, x = 2 in expression
2 (2)³ – (2)² – p (2) – 2 = 0
⇒  16 – 4 – 2p – 2 = 0
⇒  10 – 2p = 0
⇒  p = 5
(ii) Putting the value of p
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∴ 2x³ – x² – 5x – 2 = (x – 2) (2x² + 3x + 1)
The expression can be the written as
(2x³ + x² + 1) (x – 2) or (2x + 1) (x + 1) (x – 2).‍

Q4. If 𝑝(𝑥) = x³ − 6 x² + 11𝑥 − 6 & 𝑞(𝑥) = x³ − 4x² + 𝑥 + 6 , then
(i) find their L.C.M as 𝑓(𝑥).
(ii) and find their H.C.F as 𝑔(𝑥).
(iii) hence prove that: 𝑝(𝑥). 𝑞(𝑥) = 𝑓(𝑥). 𝑔(𝑥)

Explanation:
(i) For p (x), we have to find all possible factors of the constant -6
(By factor theorem, we have:) 
Factors of - 6 are ∓1, ∓2, ∓3, ∓6, etc. and on putting the value in p (x) we get:
By hit and trial: for x = 1 we get p(i) = 1³ - 6(1)² +11(1)-6 = 0.
Which clearly shows that (x-1) is a factor of p (x).
∴ on dividing p(x) by (x-1)will give:

Clearly p(x) = (x - 1)(x² - 5x + 6) which on further factorisation gives:
p(x)=(x - 1)(x - 2)(x - 3)...(A)
Similarly solving for q(x) we have:
q(x)=(x - 2)(x - 3)(x - 1)...(B)
∴ L.C.M. = (x - 1)(x - 2)(x - 3)(x + 1)...(c)

Q5. The expression 2x³ + ax² + bx – 2 leaves the remainder 7 and 0 when divided by (2x – 3) and (x + 2) respectively calculate the value of a and b. With these value of a and b factorise the expression completely.

Explanation:
Let P (x) = 2x³ + ax² + bx – 2 when P(x) is divided by 2x – 3
P\(\begin{pmatrix} \frac{3}{2} \end{pmatrix}\)=2 \(\begin{pmatrix} \frac{3}{2} \end{pmatrix}^3\)+a\(\begin{pmatrix} \frac{3}{2} \end{pmatrix}^2\)+b\(\begin{pmatrix} \frac{3}{2} \end{pmatrix}\)-2=7
=\(\frac{27}{4}\) + \(\frac{9}{4}\) a + \(\frac{3}{2}\) b - 2 = 7
= 9a + 6b = 28 + 8 – 27
= 9a + 6b = 9
= 3a + 2b = 3 …(i)
Similarly when P(x) is divided by x + 2
⇒  x = – 2
∴ 2(– 2)³ + a(– 2)² + b(– 2) – 2 = 0
⇒ – 16 + 4a – 2b – 2 = 0
⇒ 4a – 2b = 18 …(ii)
On solving equations (i) and (ii)
3a + 2b = 3
\(\underline{4a – 2b = 18}\)                (On adding (ii)              
          7a = 21          
          a = 3
On substituting value of a in equation (i)
3 × 3 + 2b = 3
2b = 3 – 9
b=\(\frac{-6}{2}\)=-3
b = – 3
a = 3, b = – 3
On substituting value of a and b
2x³ + 3x² – 3x – 2
When x + 2 is a factor

2x² – x – 1 = 2x² – 2x + x – 1
= 2x(x – 1) + 1 (x – 1)
= (x – 1) (2x + 1)
Hence, required factors are
(x – 1) (x + 2) (2x + 1)

ICSE Class 10 Maths Chapter wise Important Questions

Conclusion

The chapter factorization class 10 ICSE is a foundational and pivotal topic that serves as a cornerstone for various mathematical concepts and practical applications. Mastering these factorisation class 10 ICSE important questions equips you with valuable problem-solving skills for everyday situations, making it a crucial part of your math education.

Frequently Asked Questions

Q1 : What is factorization, and why is it important in mathematics?

Ans: Factorization is the process of expressing an algebraic expression or a number as the product of its constituent factors. It is crucial in mathematics for simplifying expressions, solving equations, and understanding mathematical relationships.

Q2: What are prime factors, and how do you find them?

Ans:  Prime factors are the smallest prime numbers that multiply together to give a given number. To find them, you can use techniques like prime factorization or factor tree diagrams.

Q3 : What is the difference between algebraic and numerical factorization?

Ans: Algebraic factorization involves breaking down algebraic expressions into their constituent factors, while numerical factorization involves finding the prime factors of a number.

Q4 : How do you factorize a quadratic trinomial like ax² + bx + c?

Ans: You can factorize quadratic trinomials using techniques like the product-sum method or trial and error to find two binomials that multiply to the trinomial.

Q5 : What is a perfect square trinomial, and how is it factorized?

Ans: A perfect square trinomial is the square of a binomial expression. It is factorized as the square of the binomial.