Page 156 - ICSE Math 8

14 Factorization of Algebraic

Expressions

Key Concepts

• Factors • Factorization of Difference of Squares
• Factorization by Taking Out Common Monomials and • Factorization of a Perfect Square Trinomial
Binomials • Factorization of Quadratic Polynomial By Splitting
• Factorization by Regrouping the Middle Term

Factorization or factoring is the process by which the factors of a composite number are determined and the
number is written as a product of these factors. Just like composite numbers, algebraic expressions also have
factors. In this chapter, we will learn about factorization of a given algebraic expression.

Factors

You know that a composite number can be written as a product of two positive integers other than 1 and the
number itself. For example, 14 is a composite number as it can be written as 7 × 2.
Similarly, an algebraic expression can also be written as a product of two or more algebraic expressions. For
2
2
example, x + 5x + 6 can be written as (x + 2) (x + 3), i.e., x + 5x + 6 = (x + 2)(x + 3).
In general, a number ‘p’ is said to be a factor of a number ‘q’ if the number p can divide the number q without
leaving any remainder. Similarly, an algebraic expression p(x) is a factor of another algebraic expression q(x)
if p(x) can divide q(x) without leaving any remainder.
For example, 8xy = 1 × 8 × x × y = (8x) × y = (8y) × x = 8 × (xy) = 1 × 8xy
So, the possible factors of 8xy are 1, 8, x, y, 8x, 8y, xy and 8xy.

Factors of a monomial
Factors of a monomial consist of literals, their product and the number that can exactly divide the monomial.
2 2
For example, the possible factors of the monomial 5x y are:
2
2
2
2 2
2
2
2
2
2 2
2 2
5x y = 1 × 5x y = 5 × x y = 5x × xy = 5xy × xy = 5y × x = 5y x × x = 5x y × y = 5y × x y = 5x × y 2
2
2
2
2 2
2 2
2
2
2
2
2
2 2
So, all the possible factors of 5x y are 1, 5x y , 5, x y , 5x, xy , 5xy, xy, 5y , x , 5y x, x, 5x y, y, 5y, x y, 5x and y .
Factors of a polynomial Try This
A polynomial can be factorized into the product of a monomial and a
2
polynomial. For example, 6x + 12x can be factorised as: 6x(x + 2), as 6x is Tick () the correct factorization
2
2
the HCF of 6x and 12x. This could be factorized in a number of ways, but of 2x – 4x.
we want it to be in a completely factorized from, i.e., a form which cannot (a) 2x(x –2)
2
(b) 2(x – 2x)
be factorized further into polynomials with integer coefficients. (c) x(2x – 4)
Highest Common Factor (HCF) of Two or More Monomials
The highest common factor of two or more monomials is the product of the common factor having greatest
numerical coefficient and the common variables with the smallest powers.
To find the HCF of two or more monomials:
Step 1: Find the HCF of the numeric coefficients of each monomial.
Step 2: Find the HCF of the common variables.
Step 3: Multiply the HCF in steps 1 and 2 to get the final HCF.
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