# NCERT Solutions Class 8 Maths Chapter 14 Exercise 14.2

The area of Mathematics known as algebra aids in the representation of circumstances or problems as mathematical expressions. To create a meaningful mathematical expression, it takes variables like x, y, and z together with mathematical operations like addition, subtraction, multiplication, and division. The students can understand its subtopics with the help of tools like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 .

Algebra is used in all areas of Mathematics, including trigonometry, calculus, and coordinate geometry. 2x + 4 = 8 is a straightforward algebraic expression.

In algebra, symbols are used, and operators are used to connect the symbols to one another. It is more than just a mathematical idea; it is a skill that students utilise on a regular basis without even being aware of it.

Understanding algebra as a concept is more crucial than correctly solving equations, since it is applicable to all other mathematical courses students will learn in the future or have already studied. Students can understand all aspects of algebra by using the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2.

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them. Variables are the name given to these symbols because they lack set values. Humans frequently observe a constant change in specific values in their day-to-day situations. But the need to depict these shifting values is ongoing.

These values are usually represented in algebra by variables, which are symbols like x, y, z, p, or q. These symbols are also subjected to various arithmetic operations—like addition, subtraction, multiplication, and division—in order to ascertain their values.

Since unknown values are known as variables, algebra is a concept based on these. Equations are a key algebraic idea. It performs arithmetic operations in accordance with a number of rules. Data sets with two or more variables can be understood using the rules. Numerous objects in our environment are analysed using it. The probability is that students will use algebra without realising it. There are several sub-branches of algebra, including commutative algebra, abstract algebra, linear algebra, and elementary algebra.

Elementary algebra or Algebra 1:

The conventional subjects covered in a contemporary primary algebra course are covered in Elementary Algebra. Arithmetic includes both numbers and mathematical operations such as +, -, and x. But in algebra, variables—such as x, a, n, and y—are frequently used to represent the integers. It is the first step that demonstrates the systematic exploration of all the properties of a system of real numbers. It also permits the common formulation of the laws of arithmetic, such as “a + b” = “b + a.” Different chapters in NCERT books cover this topic, and the students can access tools similar to those in the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, that are targeted towards this topic to understand it better.

Variables, evaluating expressions and equations, the characteristics of equalities and inequalities, solving algebraic equations, solving linear equations with one or two variables, etc. are some of the subjects covered in elementary algebra.

Advanced algebra or Algebra 2 :

This is algebra at the intermediate level. Compared to pre-algebra, this algebra has a higher level of equations to answer. Students can go through the additional algebraic concepts by using advanced algebra, such as:

- Equations involving inequality
- Matrices
- Solution to a system of linear equations
- Graphing linear equations and functions
- Curved portions
- Equation of a Polynomial
- Quadratic Inequalities in Functions
- Radical expressions and polynomials
- Series and sequences
- Reasonable statements
- Trigonometry

Abstract Algebra :

One area of algebra known as “abstract algebra” studies principles about algebraic systems, irrespective of the particulars of some operations. In particular situations, these operations have particular characteristics. As a result, certain conclusions can be drawn about these qualities. Hence, the name of this area of mathematics is “abstract algebra.”

Algebraic structures, including fields, groups, modules, rings, lattices, vector spaces, etc., are dealt with in abstract algebra.

Linear Algebra :

A division of algebra known as linear algebra has applications in both pure and applied mathematics. It deals with the vector spaces’ linear mappings. It also has to do with studying lines and planes. It is the research of linear sets of equations having properties of transformation. Almost all branches of mathematics use it. It deals with linear equations for linear functions and how they are represented in matrices and vector spaces. The following are some key areas that linear algebra covers:

- Linear equations
- Vector Spaces
- Relations
- Matrices and matrix decomposition
- Relations and Computations

Commutative Algebra :

One of the branches of algebra that analyses commutative rings and their ideals is called commutative algebra. Commutative algebra is a prerequisite for both algebraic geometry and algebraic number theory. Polynomial rings, rings of algebraic integers, and other rings are included. Commutative algebra is used in many other branches of mathematics, including differential topology, invariant theory, order theory, and general topology. It has played a noteworthy part in contemporary pure mathematics. The students can better understand this topic with the help of tools that were created by the experts in mathematics, like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, but particularly those that were created for this topic.

**NCERT Class 8 Maths Solutions, Chapter 14 Factorisation (EX 14.2) Exercise 14.2 **

In order to prevent situations where students can lose interest in a subject, students should take advantage of the resources provided by Extramarks experts. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, is just one of the many resources that students may access with ease at Extramarks.

Students can successfully complete Class 8 Maths, Chapter 14, Exercise 14.2, with the help of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and supplementary resources produced especially for Class 8 Maths, Exercise 14.2, and made available on the Extramarks website.

Examining NCERT Class 8 Maths, Chapter 14, Exercise 14.2, will help students better understand each question.

**Access NCERT Solutions for Class 8 Maths Chapter 14-Factorisation **

The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, were created by qualified mathematicians. On the Extramarks website, students can find resources for lower elementary and middle school that were produced using the same technique as the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 . Experts advise students to use materials like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, to make sure they fully comprehend everything that is taught in class and what they must write about in their exams. Students cannot comprehend the middle and high school curricula without a firm foundation in the instructional techniques utilised in elementary school classes. Students may become disinterested in a subject if they don’t have a strong foundation in it.

Students who use resources like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and others found on the Extramarks website are more likely to understand the material. They have the chance to practise the questions and learn the right technique to come to the conclusion thanks to the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 . The NCERT rules and the proper way to answer the problems were taken into consideration when creating the step-by-step solutions provided in resources like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 .

The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, can play a significant role in assisting students in effectively preparing for their exams in a subject like mathematics, where even the marks for each individual step are provided.

The Extramarks website makes it simple to find the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 . They are also simple to understand. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, are produced by the mathematics specialists at Extramarks, making them trustworthy resources as well. The mathematics specialists work to make materials like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, that are simple for everyone to understand and benefit from because the experts are aware of the needs of various students.

Students can access the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and other similar resources on the Extramarks website at any time, regardless of where they are or what time it is. The students may now plan their preparations in a very convenient way.

The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 are intended to provide students with all of the help they need to work through and practise the Chapter 14 questions. All the exercises’ questions are answered within-depth, step-by-step information for the pupils. Students may greatly improve their practise sessions and gain more confidence in their exam preparation with the aid of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and all the other NCERT solutions available on the Extramarks website.

**NCERT Class 8 Maths Solutions, Chapter 14 Factorisation Exercise 14.2 **

There is a beauty in taking a seemingly simple problem and repeatedly combining and simplifying it until there is only one value for each variable. The process can be fun, and the outcome can be quite fulfilling. That’s what Algebra is, with all its parts. That can be better understood with the help of resources like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and many more that are available on the Extramarks website.

A valuable life skill worth mastering is algebra. It takes students beyond elementary mathematics and gets them ready for calculus and statistics. It is helpful for a wide variety of vocations, some of which a student might pursue as a second career. In the home and when evaluating news articles, algebra is helpful. It also strengthens logical reasoning.

1) Algebra is more accurate and faster than “Basic” Mathematics.

Similar to how multiplication by 12 is quicker than counting to 24 or adding 2 by 12, algebra makes problem-solving quicker and simpler than it would be without it. Additionally, algebra enables the solution of previously unsolvable life problems like charting curves that go beyond the scope of basic math abilities. Chapter 14 teaches factorization, which is a part of Algebra, and the students can understand it better with the help of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2

2) Students need Algebra to understand Statistics and Calculus, and they can do so with the help of resources available on the Extramarks website like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and more.

Calculus and statistics are two types of mathematics that many people utilise in their careers, even though the idea of learning one type of mathematics in order to acquire more types of arithmetic may not be immediately rewarding. Statistics are generally employed in a variety of positions in the business world, the media, the health and wellness industry, politics, the social sciences, and many other sectors. Understanding statistics helps students become better workers, citizens, and information consumers. This can be achieved with a tool similar to the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 .

3) Numerous intricate processes, such as how an object’s speed changes over time, are better understood because of calculus. Calculus is a tool that engineers and scientists use to create consumer goods, medicinal procedures, and new technologies. Any student interested in a job in science, medicine, computer modelling, or engineering must learn calculus. They can start to form the base by making use of all the related resources provided on the Extramarks website, like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 .

4) Later on, knowing algebra could be a useful skill.

Even if a student is certain that they will not be entering a profession that requires statistics or calculus, many people change professions and entire careers throughout their working lives, and factorization and other parts of algebra may play an important role at some point or another. The students are advised to gain a strong foundation in this topic with the help of tools like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2. and more.

Factorization is the process of breaking down a number or polynomial into a product of numerous factors of other polynomials that, when multiplied, produce the original number. Use the factorization formula to divide an integer into its factors. A number, matrix, or polynomial can be converted into the product of another entity or factors, which, when multiplied together, produce the original number. This process is known as “factorization.” A large number is split up into smaller numbers, or factors, using the factorization formula. A factor is a number that evenly divides an integer by itself with no remainder. Prime Factorization of 28 is, for instance, 2 X 2 X 7, and before we begin factorization, let’s first go through the mathematical concept of “factor.” The students are advised to make the most of resources like the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, if they wish to master this topic. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, can prove to be very beneficial in helping the students gain in-depth knowledge of the topic. The Solutions for Class 8 Maths Exercise 14.2 are prepared by the experts in the topic. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, are therefore essential for the students to better understand this particular chapter.

Finding the factors of an algebraic expression, also known as locating two or more expressions whose product is the given expression, is the process of factorization. The factorization of algebraic expressions is the process of identifying two or more expressions whose product is the given expression. A factor is a number that evenly divides the inputted number. It merely refers to representing a number as the product of two other numbers. Similar to this, algebraic expressions are written as the product of their factors. The only exception is that an algebraic expression here combines addition or subtraction with arithmetic operations like numbers and variables. The students can find this to be a challenging topic, but they are advised by the experts to make use of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, to get a better understanding of the topic.

The students can make use of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, to help them feel less intimidated by the topic of factorization. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, have step-by-step solutions to all the NCERT questions. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 can really be useful to practise and master the themes covered in Chapter 14 and specifically Exercise 14.2 .

**NCERT Solutions for Class 8 **

A number is expressed as the product of any two numbers using the term “factor.” Finding factors for any mathematical object, such as an integer, polynomial, or algebraic expression, is done through the process of factorization. Finding the factors of the provided algebraic expression is what is meant when an algebraic expression is factorised. The students can practise this topic and understand it better with the help of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and other tools on the Extramarks website that are targeted towards this topic.

There are numerous ways to factorise algebraic expressions. The most typical techniques for factoring algebraic expressions include:

- Using common factors to factorise.
- Factorization by term regrouping.
- Utilizing identities to factorise.

Mathematicians frequently employ the procedure of factoring. This aids in resolving various mathematical issues or in the simplification of formulas. A polynomial is divided into two or more factors through factoring. Students have to keep in mind that the original phrase should still be obtained when they multiply these components.

Numerous daily tasks include the use of factors. Students are aware that factoring makes it possible to divide something into several equal parts, so the concept of factoring is present in any division into equal parts.

Finding the dimensions of a specific area, such as a pool, backyard, or other space, is another example of factoring. Students can notice that factoring is an integral part of our everyday lives. To help make these real life situations easier, the students should try to master this topic with the help of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and other similar tools. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2 are easily accessible and simple to understand. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, are prepared by experts in mathematics who have good knowledge of the topic of factorization.

Factorization can be a very interesting topic for the students to study if they try to understand and study the real-life uses of it. The students can do so with the help of tools available on the Extramarks website, including the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and more. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, are just one set of tools created for Chapter 14 of Class 8 Mathematics.

Preparing for topics such as Factorization or anything related to Algebra can make students form the preconceived notion that it is going to be too complex and boring. If the students regularly access the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and other such tools available on the Extramarks website, they can avoid this situation. It is common knowledge that each class establishes some significant conceptual groundwork for later study. The eighth grade is one such critical stepping stone where students discover many fresh and vital ideas and principles that will be very helpful in their future academic endeavours. Because of this, parents and teachers frequently put pressure on the students to focus more and take their academics more seriously. The students can reduce the stress by practising more. They can do so with the help of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and other similar tools. According to the CBSE curriculum, the mathematics textbook for Class 8 has sixteen chapters, one of which is Factorization. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, and other resources geared towards Chapter 14 of Class 8 Mathematics can help you better understand and practise factorization.

The use of the NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, not only helps with final examination preparations but also with the homework that students get on a regular basis. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, can also help students who wish to prepare their class presentation on this topic. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, can be a great tool to help students understand the topic so well that they can work on creating a project on it if and when they need to.

Students are frequently expected to work on such activities as it increases their understanding of individual topics and also makes the process of studying mathematics fun.

The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, is a tool that can help students understand the topic well enough to actively participate in the class activities. The class activities are not only fun but also part of the guidelines issued to the schools. The schools are asked to conduct these activities to help the students better understand the topics, and they are mandatory to participate in. The marks from these activities and class tests are added to the final results under the ”internal marking” section. The NCERT Solutions for Class 8 Maths, Chapter 14, Exercise 14.2, can help students score well on their internal assessments.

**Q.1 **

$\begin{array}{l}\text{Factorise the following expressions.}\\ {\text{(i) a}}^{\text{2}}\text{+ 8a + 16}\\ {\text{(ii) p}}^{\text{2}}-\text{10 p + 25}\\ {\text{(iii) 25m}}^{\text{2}}\text{+ 30m + 9}\\ {\text{(iv) 49y}}^{\text{2}}{\text{+ 84yz + 36z}}^{\text{2}}\text{}\\ {\text{(v) 4x}}^{\text{2}}-\text{8x + 4}\\ {\text{(vi) 121b}}^{\text{2}}-{\text{88bc + 16c}}^{\text{2}}\\ {\text{(vii) (l + m)}}^{\text{2}}-{\text{4lm (Hint: Expand ( l + m)}}^{\text{2}}\text{first)}\\ {\text{(viii) a}}^{\text{4}}{\text{+ 2a}}^{\text{2}}{\text{b}}^{\text{2}}{\text{+ b}}^{\text{4}}\end{array}$

**Ans**

(i) a^{2} + 8a + 16

The given expression can be written as:

= (a)^{2} + 2 × a × 4 + (4)^{2}

This is of the form: x^{2}+ y^{2 }+2xy with x = a, y = 4 and 2xy = 8a

Therefore, by using the identity: (x + y)^{2} = x^{2} + 2xy + y^{2},we get

= (a + 4)^{2}

(ii) p^{2} − 10p + 25

The given expression can be written as:

= (p)^{2} − 2 × p × 5 + (5)^{2}

This is of the form: a^{2}+ b^{2 }−2ab with a= p, b= 5 and

2ab =10p

Therefore, by using the identity: (a − b)^{2} = a^{2} – 2ab + b^{2},we get

= (p − 5)^{2}

(iii) 25m^{2} + 30m + 9

The given expression can be written as:

= (5m)^{2} + 2 × 5m × 3 + (3)^{2}

This is of the form: a^{2}+ b^{2 }+2ab with a = 5m, b = 3 and 2ab =30m

Therefore, by using the identity: (a + b)^{2} = a^{2} + 2ab + b^{2},we get

= (5m + 3)^{2}

(iv) 49y^{2 }+ 84yz + 36z^{2}

The given expression can be written as:

= (7y)^{2} + 2 × (7y) × (6z) + (6z)^{2}

This is of the form: a^{2 }+ b^{2 }+2ab with a = 7y, b= 6z and 2ab = 84yz

Therefore, by using the identity: (a + b)^{2} = a^{2} + 2ab + b^{2},we get

= (7y + 6z)^{2}

(v) 4x^{2 }− 8x + 4

The given expression can be written as:

= (2x)^{2} − 2 (2x) (2) + (2)^{2}

This is of the form: a^{2 }+ b^{2 }− 2ab with a = 2x, b= 2 and 2ab = 8x

Therefore, by using the identity: (a − b)^{2} = a^{2} − 2ab + b^{2},we get

= (2x − 2)^{2}

Again, we can take out 2 common

= [(2) (x − 1)]^{2}

= 4(x − 1)^{2}

(vi) 121b^{2 }− 88bc + 16c^{2}

The given expression can be written as:

= (11b)^{2} − 2 (11b) (4c) + (4c)^{2}

This is of the form: a^{2 }+ b^{2 }− 2ab with a = 11b, b= 4c and 2ab = 88bc

Therefore, by using the identity: (a − b)^{2} = a^{2} − 2ab + b^{2},we get

= (11b − 4c)^{2}

(vii) (l + m)^{2} − 4lm

The given expression can be written as:

= l^{2} + 2lm + m^{2} − 4lm

= l^{2} − 2lm + m^{2}

This is of the form: a^{2 }+ b^{2 }− 2ab with a = l, b= m and 2ab = 2lm

Therefore, by using the identity: (a − b)^{2} = a^{2} − 2ab + b^{2},we get

= (l − m)^{2}

(viii) a^{4} + 2a^{2}b^{2} + b^{4}

The given expression can be written as:

= (a^{2})^{2} + 2 (a^{2}) (b^{2}) + (b^{2})^{2}

This is of the form: x^{2 }+ y^{2 }+ 2xy with x = a^{2}, y= b^{2} and 2xy = 2a^{2}b^{2}

Therefore, by using the identity: (x + y)^{2} = x^{2} + 2xy + y^{2},we get

= (a^{2} + b^{2})^{2}

**Q.2 **

$\begin{array}{l}\text{Factorise.}\\ {\text{(i)\hspace{0.33em}4p}}^{\text{2}}\text{\hspace{0.33em}}-{\text{\hspace{0.33em}9q}}^{\text{2}}\text{}\\ {\text{(ii)\hspace{0.33em}63a}}^{\text{2}}-{\text{\hspace{0.33em}112b}}^{\text{2}}\text{}\\ {\text{(iii)\hspace{0.33em}49x}}^{\text{2}}\text{\hspace{0.33em}}-\text{36}\\ {\text{(iv)\hspace{0.33em}16x}}^{\text{5}}\text{\hspace{0.33em}}-{\text{\hspace{0.33em}144x}}^{\text{3}}\text{}\\ {\text{(v)\hspace{0.33em}(l + m)}}^{\text{2}}\text{\hspace{0.33em}}-\text{\hspace{0.33em}}{\left(\text{l}-\text{\hspace{0.33em}m}\right)}^{\text{2}}\\ {\text{(vi)\hspace{0.33em}9x}}^{\text{2}}{\text{y}}^{\text{2}}\text{\hspace{0.33em}}-\text{\hspace{0.33em}16}\\ {\text{(vii)\hspace{0.33em}(x}}^{\text{2}}-{\text{\hspace{0.33em}2xy + y}}^{\text{2}}\text{)}-{\text{\hspace{0.33em}z}}^{\text{2}}\\ {\text{(viii)\hspace{0.33em}25a}}^{\text{2}}-{\text{\hspace{0.33em}4b}}^{\text{2}}\text{+ 28bc}-{\text{\hspace{0.33em}49c}}^{\text{2}}\end{array}$

**Ans**

(i) 4p^{2} − 9q^{2}

The given expression can be written as:

= (2p)^{2} − (3q)^{2}

This is of the form: a^{2} − b^{2 }with a = 2p and b = 3q

Therefore, by using the identity: a^{2} − b^{2} = (a − b) (a + b), we get

= (2p + 3q) (2p − 3q)

(ii) 63a^{2} − 112b^{2}

The given expression can be written as:

= 7(9a^{2} − 16b^{2})

= 7[(3a)^{2} − (4b)^{2}]

This is of the form: x^{2} − y^{2 }with x = 3a and y = 4b

Therefore, by using the identity: x^{2} − y^{2} = (x − y) (x + y), we get

= 7(3a + 4b) (3a − 4b)

(iii) 49x^{2} − 36

The given expression can be written as:

= (7x)^{2} − (6)^{2}

= (7x − 6) (7x + 6)

(iv) 16x^{5} − 144x^{3}

The given expression can be written as:

= 16x^{3}(x^{2} − 9)

= 16 x^{3} [(x)^{2} − (3)^{2}]

This is of the form: a^{2} − b^{2 }with a = x and b = 3

Therefore, by using the identity: a^{2} − b^{2} = (a − b)(a + b), we get

= 16x^{3}(x − 3) (x + 3)

(v) (l + m)^{2} − (l − m)^{2}

This is of the form: a^{2} − b^{2 }with a = (l + m) and

b = (l − m)

Therefore, by using the identity: a^{2} − b^{2} = (a − b)(a + b), we get

= [(l + m) − (l − m)] [(l + m) + (l − m)]

= (l + m − l + m) (l + m + l − m)

= 2m × 2l

= 4ml

= 4lm

(vi) 9x^{2}y^{2} − 16

The given expression can be written as:

= (3xy)^{2} − (4)^{2}

This is of the form: a^{2} − b^{2 }with a = 3xy and b = 4

Therefore, by using the identity: a^{2} − b^{2} = (a − b)(a + b), we get

= (3xy − 4) (3xy + 4)

(vii) (x^{2} − 2xy + y^{2}) − z^{2}

By using the identity: (a − b)^{2} = a^{2} − 2ab + b^{2},we get

= (x − y)^{2} − (z)^{2}

This is of the form: a^{2} − b^{2 }with a = (x – y) and b = z

Therefore, by using the identity: a^{2} − b^{2} = (a − b)(a + b), we get

= (x − y − z) (x − y + z)

(viii) 25a^{2} − 4b^{2} + 28bc − 49c^{2}

The given expression can be written as:

= 25a^{2} − (4b^{2} − 28bc + 49c^{2})

= (5a)^{2} − [(2b)^{2} − 2 × 2b × 7c + (7c)^{2}]

By using the identity: (a − b)^{2} = a^{2} − 2ab + b^{2},we get

= (5a)^{2} − [(2b − 7c)^{2}]

This is of the form: x^{2} − y^{2 }with x = 5a and y = (2b – 7c)

Therefore, by using the identity: x^{2} − y^{2} = (x − y)(x + y), we get

= [5a + (2b − 7c)] [5a − (2b − 7c)]

= (5a + 2b − 7c) (5a − 2b + 7c)

**Q.3 **

$\begin{array}{l}\text{Factorise the expressions.}\\ {\text{(i)\hspace{0.33em}ax}}^{\text{2}}\text{+ bx}\\ {\text{(ii)\hspace{0.33em}7p}}^{\text{2}}{\text{+ 21q}}^{\text{2}}\text{}\\ {\text{(iii)\hspace{0.33em}2x}}^{\text{3}}{\text{+ 2xy}}^{\text{2}}{\text{+ 2xz}}^{\text{2}}\\ {\text{(iv)\hspace{0.33em}am}}^{\text{2}}{\text{+ bm}}^{\text{2}}{\text{+ bn}}^{\text{2}}{\text{+ an}}^{\text{2}}\text{}\\ \text{(v)\hspace{0.33em}(lm + l) + m + 1}\\ \text{(vi)\hspace{0.33em}y (y + z) + 9 (y + z)}\\ {\text{(vii)\hspace{0.33em}5y}}^{\text{2}}\text{\hspace{0.33em}}-\text{\hspace{0.33em}20y\hspace{0.33em}}-\text{\hspace{0.33em}8z + 2yz}\\ \text{(viii)\hspace{0.33em}10ab + 4a + 5b + 2}\\ \text{(ix)\hspace{0.33em}6xy\hspace{0.33em}}-\text{\hspace{0.33em}4y + 6\hspace{0.33em}}-\text{\hspace{0.33em}9x}\end{array}$

**Ans**

(i) ax^{2} + bx

= a × x × x + b × x

Here, the common factor is ‘x’.

Therefore,

ax^{2} + bx = x(ax + b)

(ii) 7p^{2} + 21q^{2}

= 7 × p × p + 3 × 7 × q × q

Here, the common factor is ‘7’.

Therefore,

7p^{2} + 21q^{2} = 7(p^{2} + 3q^{2})

(iii) 2x^{3} + 2xy^{2} + 2xz^{2}

Here, the common factor is ‘2x’.

Therefore,

2x^{3} + 2xy^{2} + 2xz^{2} = 2x(x^{2} + y^{2} + z^{2})

(iv) am^{2} + bm^{2} + bn^{2} + an^{2}

The given expression can be written as:

= am^{2} + bm^{2} + an^{2 }+ bn^{2}

The common factor in first two terms is m^{2} and the common factor in last two terms is n^{2}.

= m^{2}(a + b) + n^{2}(a + b)

= (a + b) (m^{2} + n^{2})

Therefore,

am^{2} + bm^{2} + bn^{2} + an^{2} = (a + b) (m^{2} + n^{2})

(v) (lm + l) + m + 1

Rearranging the given expression as follows:

= lm + m + l + 1

Taking out common factors .

= m(l + 1) + 1(l + 1)

= (l + 1) (m + 1)

Therefore,

(lm + l) + m + 1=(l + 1) (m + 1)

(vi) y (y + z) + 9 (y + z)

= (y + z) (y + 9)

Therefore,

y (y + z) + 9 (y + z) = (y + z) (y + 9)

(vii) 5y^{2} − 20y − 8z + 2yz

The given expression can be written as follows:

5y^{2} − 20y + 2yz − 8z

Taking out common factors.

= 5y(y − 4) + 2z(y − 4)

= (y − 4) (5y + 2z)

Therefore,

5y^{2} − 20y − 8z + 2yz =(y − 4) (5y + 2z)

(viii) 10ab + 4a + 5b + 2

Rearranging the given expression as follows:

= 10ab + 5b + 4a + 2

Taking out the common terms

= 5b(2a + 1) + 2(2a + 1)

= (2a + 1) (5b + 2)

Therefore,

10ab + 4a + 5b + 2= (2a + 1) (5b + 2)

(ix) 6xy − 4y + 6 − 9x

Rearranging the terms of the given expression and taking out common terms.

= 6xy − 9x − 4y + 6

= 3x(2y − 3) − 2(2y − 3)

= (2y − 3) (3x − 2)

Therefore,

6xy − 4y + 6 − 9x= (2y − 3) (3x − 2)

**Q.4 **

$\begin{array}{l}\text{Factorise.}\\ {\text{(i)\hspace{0.33em}a}}^{\text{4}}-{\text{\hspace{0.33em}b}}^{\text{4}}\text{}\\ {\text{(ii)\hspace{0.33em}p}}^{\text{4}}-\text{\hspace{0.33em}81}\\ {\text{(iii)\hspace{0.33em}x}}^{\text{4}}-{\text{\hspace{0.33em}(y + z)}}^{\text{4}}\\ {\text{(iv)\hspace{0.33em}x}}^{\text{4}}-{\text{(x \u2013 z)}}^{\text{4}}\text{}\\ {\text{(v)\hspace{0.33em}a}}^{\text{4}}-{\text{2a}}^{\text{2}}{\text{b}}^{\text{2}}\text{+ b4}\end{array}$

**Ans**

$\begin{array}{l}\left(\mathrm{i}\right)\text{\hspace{0.33em}}\mathrm{}{\mathrm{a}}^{4}-{\mathrm{b}}^{4}\\ =\text{}{\left({\mathrm{a}}^{2}\right)}^{2}-\text{}{\left({\mathrm{b}}^{2}\right)}^{2}\\ =\text{}({\mathrm{a}}^{2}-{\mathrm{b}}^{2})\text{}({\mathrm{a}}^{2}+{\mathrm{b}}^{2})\text{}\left[\mathrm{By}\text{}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ =\text{}(\mathrm{a}-\mathrm{b})\text{}(\mathrm{a}+\mathrm{b})\text{}({\mathrm{a}}^{2}+{\mathrm{b}}^{2})\left[\mathrm{By}\text{}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ \\ \left(\mathrm{ii}\right)\text{\hspace{0.33em}}{\mathrm{p}}^{4}-\text{}81\text{}\\ ={\left({\mathrm{p}}^{2}\right)}^{2}-\text{}{\left(9\right)}^{2}\\ =({\mathrm{p}}^{2}-\text{}9)\text{}({\mathrm{p}}^{2}+\text{}9)\left[\mathrm{By}\text{}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ =[{\left(\mathrm{p}\right)}^{2}-\text{}{\left(3\right)}^{2}]\text{}({\mathrm{p}}^{2}+\text{}9)\\ =(\mathrm{p}-\text{}3)\text{}(\mathrm{p}+\text{}3)\text{}({\mathrm{p}}^{2}+\text{}9)\left[\mathrm{By}\text{}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ \\ \left(\mathrm{iii}\right)\text{\hspace{0.33em}}{\mathrm{x}}^{4}-{(\mathrm{y}+\mathrm{z})}^{4}\\ ={\left({\mathrm{x}}^{2}\right)}^{2}-{\left[{(\mathrm{y}+\mathrm{z})}^{2}\right]}^{2}\\ =[{\mathrm{x}}^{2}-{(\mathrm{y}+\mathrm{z})}^{2}][{\mathrm{x}}^{2}+{(\mathrm{y}+\mathrm{z})}^{2}]\text{}\left[\mathrm{By}\text{}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\end{array}$ $\begin{array}{l}=[\mathrm{x}-(\mathrm{y}+\mathrm{z})][\mathrm{x}+(\mathrm{y}+\mathrm{z})][{\mathrm{x}}^{2}+{(\mathrm{y}+\mathrm{z})}^{2}]\\ \text{}\left[\mathrm{By}\text{\hspace{0.33em}}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ =(\mathrm{x}-\mathrm{y}-\mathrm{z})\text{}(\mathrm{x}+\mathrm{y}+\mathrm{z})\text{}[{\mathrm{x}}^{2}+{(\mathrm{y}+\mathrm{z})}^{2}]\\ \\ \left(\mathrm{iv}\right)\text{\hspace{0.33em}}{\mathrm{x}}^{\text{4}}-{(\mathrm{x}-\mathrm{z})}^{\text{4}}\\ ={\left({\mathrm{x}}^{\text{2}}\right)}^{\text{2}}-{\left[{(\mathrm{x}-\mathrm{z})}^{\text{2}}\right]}^{\text{2}}\\ =[{\mathrm{x}}^{\text{2}}-{(\mathrm{x}-\mathrm{z})}^{\text{2}}][{\mathrm{x}}^{\text{2}}+{(\mathrm{x}-\mathrm{z})}^{\text{2}}]\text{}\left[\mathrm{By}\text{\hspace{0.33em}}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ =[\mathrm{x}-(\mathrm{x}-\mathrm{z})]\text{}[\mathrm{x}+(\mathrm{x}-\mathrm{z})][{\mathrm{x}}^{\text{2}}+{(\mathrm{x}-\mathrm{z})}^{\text{2}}]\\ \text{}\left[\mathrm{By}\text{\hspace{0.33em}}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ =\mathrm{z}(\text{2}\mathrm{x}-\mathrm{z})[{\mathrm{x}}^{\text{2}}+{\mathrm{x}}^{\text{2}}-\text{2}\mathrm{xz}+{\mathrm{z}}^{\text{2}}]\\ =\mathrm{z}(\text{2}\mathrm{x}-\mathrm{z})\text{}(\text{2}{\mathrm{x}}^{\text{2}}-\text{2}\mathrm{xz}+{\mathrm{z}}^{\text{2}})\\ \\ \left(\mathrm{v}\right)\text{\hspace{0.33em}}{\mathrm{a}}^{\text{4}}-\text{2}{\mathrm{a}}^{\text{2}}{\mathrm{b}}^{\text{2}}+{\mathrm{b}}^{\text{4}}\\ ={\left({\mathrm{a}}^{\text{2}}\right)}^{\text{2}}-\text{2}\left({\mathrm{a}}^{\text{2}}\right)\left({\mathrm{b}}^{\text{2}}\right)+{\left({\mathrm{b}}^{\text{2}}\right)}^{\text{2}}\\ ={({\mathrm{a}}^{\text{2}}-{\mathrm{b}}^{\text{2}})}^{\text{2}}\\ ={\left[(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b})\right]}^{\text{2}}\text{\hspace{0.17em}}\left[\mathrm{By}\text{\hspace{0.33em}}\mathrm{using}:({\mathrm{a}}^{2}-{\mathrm{b}}^{2})=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right]\\ ={(\mathrm{a}-\mathrm{b})}^{\text{2}}{(\mathrm{a}+\mathrm{b})}^{\text{2}}\end{array}$

**Q.5 **

$\begin{array}{l}\text{Factorise the following expressions.}\\ {\text{(i) p}}^{\text{2}}\text{+ 6p + 8}\\ {\text{(ii) q}}^{\text{2}}-\text{10q + 21}\\ {\text{(iii) p}}^{\text{2}}\text{+ 6p}-\text{16}\end{array}$

**Ans**

(i) p^{2} + 6p + 8

Here, 8 = 4 × 2 and

6 = 4 + 2

Therefore, p^{2} + 6p + 8 can be written as:

p^{2} + 2p + 4p + 8

= p(p + 2) + 4(p + 2)

= (p + 2) (p + 4)

(ii) q^{2} − 10q + 21

Here, 21 = (−7) × (−3) and

−10 = (−7) + (−3)

Therefore, q^{2} − 10q + 21 can be written as:

q^{2} − 7q − 3q + 21

= q(q − 7) − 3(q − 7)

= (q − 7) (q − 3)

(iii) p^{2} + 6p − 16

Here, 16 = (−2) × 8 and

6 = 8 + (−2)

Therefore, p^{2} + 6p – 16 can be written as:

= p^{2} + 8p − 2p − 16

= p(p + 8) − 2(p + 8)

= (p + 8) (p – 2)

##### FAQs (Frequently Asked Questions)

## 1. What type of questions are there in the Chapter 14 of Class 8 Mathematics?

There are questions about algebraic expressions and natural numbers in chapter 14 on factorization. This chapter discusses significant issues such typical factorization methods, factorization utilizing identities, factorization by regrouping words, factors of the form (x + a) (x + b), and division of algebraic expressions. The NCERT Solutions for Class 8 Maths Chapter 14 Exercise 14.2 can help students solve these questions.

The NCERT textbook contains 4 exercises and 34 questions. These questions are all answered in detail in the NCERT Solutions for Class 8 Maths Chapter 14 Exercise 14.2 . Following each topic is some practise questions to aid with grasping the concept. In the NCERT Solutions for Class 8 Maths Chapter 14 Exercise 14.2, students may also find answers to these practise questions.