Factorisation Class 8 Maths Notes - Chapter 14

Factorisation

Algebraic expressions are the combinations of variables and constants with some relation between them.
For example: 99abcd, 2a + 6b – 11c + 9ab

Let’s see the factors of 2pq.

2pq = 2 × p × q

As we can see that the factors cannot be further simplified so, 2, p, and q are the prime factors of 2pq. In the case of factorisation of algebraic expressions, we use the word ‘irreducible form’ instead of prime factors.

Let’s learn to do factorisation by different methods.

Common Factors

In this method, we use common factors to factorise the algebraic expressions. First, factorise all the terms. Then, take the common factor out of the terms to factorise.

Let’s understand this by an example.

Factorise: 39abc – 52a2b2

The prime factorisation of 39abc and 52a2b2 is given as:

39abc = 3 × 13 × a × b × c
52a2b2 = 2 × 2 × 13 × a × a × b × b

We can see that 13, a, and b are the common factors. Therefore,

39abc – 52a2b2 = 3 × 13 × a × b × c – 2 × 2 × 13 × a × a × b × b
= 13 × a × b (3 × c – 2 × 2 × a × b)
= 13ab (3c – 4ab)

Regrouping

In regrouping, we collect the like terms, that is, the algebraic terms together which have some common factors and are easy to factorise.

Let’s understand it by an example:

Factorise: pq2 − qn2 − pq + n2

pq2 − qn2 − pq + n2
=(pq2 − pq) + (n2 − qn2) [Regroup the terms]
= pq(q − 1) + n2(1 − q)
=pq(q − 1) − n2(q − 1) [∵(1 − q)= −(q − 1)]
=(pq − n2)(q − 1) [Taking (q − 1) as the common factor]

Let us see how we can use identities for factorisation.

NCERT Solutions for Class 8 Maths Chapter 14
NCERT Solutions Class 8 Maths Chapter 14
NCERT Solutions Class 8 Maths Exercise 14.1
NCERT Solutions Class 8 Maths Exercise 14.2
NCERT Solutions Class 8 Maths Exercise 14.3
NCERT Solutions Class 8 Maths Exercise 14.4

Factorisation Using Identities

The three important and generally used algebraic identities are:

(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(a2 – b2) = (a + b)(a – b)

We use these identities in the factorisation of algebraic expressions.

Examples:

Factorise: 4y2 + 12yz + 9z2

Rewrite the algebraic expression in the form of identity.
4y2 + 12yz + 9z2 = (2y)2 + 2 (2y) (3z) + (3z)2
Use the identity (a + b)2 = a2 + 2ab + b2,
4y2 + 12yz + 9z2 = (2y + 3z)2
= (2y + 3z) (2y + 3z)

Factors of the form (x + a) (x + b)

Consider an algebraic expression px2 + qx + r. We need to find the values of two factors, a and b, such that a + b = q and ab = pr. After that, write the algebraic expression in the form of (a + b) and ab to factorise it.

Let’s understand by an example.

x2 + 14x + 45 is of the form px2 + qx + r.
p = 1, q = 14 and r = 45
We find a and b such that ab = p × r = 1 × 45 = 45 and a + b = q = 14
9 × 5 = 45
9 + 5 = 14

We get a = 9 and b = 5.

Rewrite the algebraic expression.

x2 + 14x + 45 = x2 + 9x + 5x + 45
= x (x + 9) + 5 (x + 9)
= (x + 9) (x + 5)

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Division of Algebraic Expressions

We have learnt how to add, subtract and multiply algebraic expressions. Let's see the division of different types of polynomials.

Monomial by Monomial

Let’s take an example to understand this.

Divide: 310xyz ÷ 62x2y

310xyz = 2 × 5 × 31 × x × y × z
62x2y = 2 × 31 × x × x × y

Write in division form and cancel the common factors.

310xyz ÷ 62x2y

Polynomial by Monomial

Divide yx3 + 15x2 – 77xyz by 11y.

Polynomial by Polynomial

Solve: (x + y)2 – 4xy – 16z2 ÷ (x – y – 4z)

(x + y)2 – 4xy – 16z2 = x2 + 2xy + y2 – 4xy – 16z2
= (x2 – 2xy + y2) – 16z2
= (x – y)2 – (4z)2
= (x – y + 4z) (x – y – 4z)

Write in the division form and cancel out the common factor.

(x + y)2 – 4xy – 16z2 ÷ (x – y – 4z)



Now, we’ll move on to an interesting section.

Error Finding

While solving equations, many times it happens that we do some errors but it’s difficult to find out as the equations look correct. Let us understand this with the help of examples.

Is the solution correct?

(5x)2 + 6x
= 25x + 6x
= 31x

No, the solution is not correct.

The correct solution should be:

(5x)2 + 6x
= 25x2 + 6x

To find whether the given equation/solution is correct or not, just solve any side of the equation and see if you get the expression on the other side as the answer.

Links to Other Resources for Class 8 Maths

Other Class 8 Maths Related Links
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