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Points to Remember- Factorisation Notes | Study Mathematics (Maths) Class 8 - Class 8

Document Description: Points to Remember- Factorisation for Class 8 2022 is part of Mathematics (Maths) Class 8 preparation. The notes and questions for Points to Remember- Factorisation have been prepared according to the Class 8 exam syllabus. Information about Points to Remember- Factorisation covers topics like FACTORISATION USING COMMON FACTORS, FACTORISATION BY REGROUPING TERMS, Solved Examples: and Points to Remember- Factorisation Example, for Class 8 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Points to Remember- Factorisation.

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Table of contents
FACTORISATION USING COMMON FACTORS
FACTORISATION BY REGROUPING TERMS
Solved Examples:
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• Factorisation means to write an expression as a product of its factors.
• Prime factor, an irreducible factor, a factor which cannot be expressed further as a product of factors.
• Some expressions can easily be factorised using these identities:
I. a2 + 2ab + b2 = (a + b)2
II. a2 – 2ab + b2 = (a – b)2
III. a2 – b2 = (a – b)(a + b)
IV. x2 + (a + b)x + ab = (x + a)( x+ b)
• The number 1 is a factor of every algebraic term, but it is shown only when needed.
• When factorisation of x2 + (a + b)x + ab is done by splitting the middle term, the two numbers which give the product ab and (a + b) as the coefficient of x have to be chosen very carefully with correct sign.

We Know That
(i) (a + b)2 = a2 + b2 + 2ab
(ii) (a – b)2 = a2 + b2 – 2ab
(iii) a2 – b2 = (a + b)(a – b)
(iv) 1 is a factor of every term of an algebraic expression. Unless it is specially required, we do not show 1 as a separate factor of any term.
(v) Factorisation means writing an expression as product of factors.


Note: 
In case of factorisation of a term of an expression, the word ‘irreducible’ is used in place of ‘prime’. For example, 6pq = 2 * 3 * pq is not the irreducible form because pq can further be factorised as p q, i.e. the irreducible form of 6pq = 2 * 3 * p * q.

Example 1: Write 10y as irreducible factor form.
Solution:
We have  
10 = 2 * 5
xy = x *y
∴  10xy = 2 * 5* x * y

FACTORISATION USING COMMON FACTORSExample 1: Factorise 18x2 – 14x3 + 10x4
Solution:
We have  
Points to Remember- Factorisation Notes | Study Mathematics (Maths) Class 8 - Class 8
Obviously, the common factors of these terms are 2, and .
∴ 18x2 – 14x+ 10x4
Points to Remember- Factorisation Notes | Study Mathematics (Maths) Class 8 - Class 8
Thus, 18x2 – 14x3 + 10x4 = 2x2[9 – 7x + 5x2]

FACTORISATION BY REGROUPING TERMSIn certain cases the given expression cannot be factorised easily but by rearranging its terms, we can form groups leading to factorization.
Eample 1: Factorise 9x + 18y + 6xy + 27
Solution: 
Here, we have a common factor 3 in all the terms.
∴ 9x + 18y + 6xy + 27 = 3[3x + 6y + 2xy + 9]
We find that 3x + 6y = 3(x + 2y) and 2xy + 9 = 1(2xy + 9)
i.e. a common factor in both the groups does not eist,
Thus, 3x + 6y + 2xy + 9 cannot be factorised.
On regrouping the terms, we have
3x + 6y + 2xy + 9 = 3x + 9 + 2xy + 6y
= 3(x + 3) + 2y(x + 3)
= (x + 3)(3 + 2y)
Now, 3[3x + 6y + 2xy + 9] = 3[(x + 3)(3 + 2y)]
Thus, 9x + 18y + 6xy + 27 = 3(x + 3)(2y + 3)

Solved Examples:
Q1: Let f(x)=2x3+16x2+44x+42 be a polynomial having one of the factors as (x2+5x+7), then the other factor of f(x) would be a multiple of:
A) 1
B) 2
C) 3
D) 4
Solution:
B) Since f(x) is a cubic polynomial, and one of the factors is a polynomial of degree two, then we can say that the other factor will be a polynomial of the form ax + b; where ‘a’ nd ‘b’ are two constants and a ≠ 0. Hence, we can write:
2x3+16x2+44x+42 = (x2+5x+7) × (ax + b) = ax3 + bx2 + 5ax2 + 5bx + 7ax + 7b
or 2x3+16x2+44x+42 = ax3 + (b + 5a)x2 +x2 + (5b + 7a)x + 7bCompairing the coefficients of x on both sides, we have 2 = a and 42 = 7b. Therefore, b = 6 and a = 2. hence the other factor is 2x + 6 or 2(x+3) which is a multiple of 2.

Q 2: Factorise: 5m2 − 8m − 4:
A) (5m + 2)(m + 2)
B) (5m – 2)(m – 2)
C) (5m – 2)(m + 2)
D) (5m + 2)(m – 2)
Solution:
D) The given expression is: 5m2 – 8m – 4. Therefore, it can be written as:
5m2 – 10m + 2m – 4 = 5m(m – 2) + 2(m –  2)
Hence we can write this = (5m + 2)(m – 2)
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