Page 124 - Start Up Mathematics_8 (Non CCE)

7 Factorization of Algebraic Expressions

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 is a number that can be written as a product of two positive integers other
than 1 and the number itself. For example, 14 is a composite number because it can be written as 7 times 2
(i.e., 14 = 7 × 2).

A composite algebraic expression is similar as it can be written as a product of two or more algebraic
2
expressions. For example, x + 5x + 6 is a composite algebraic expression because it can be written as
2
(x + 2) (x + 3), i.e., x + 5x + 6 = (x + 2)(x + 3).
In general, a number ‘p’ is a factor of a number ‘q’ if the number p can divide the number q without 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 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 divide the monomial exactly.
2 2
For example, find all the possible factors of the monomial 5x y .
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 .
Highest Common Factor (HCF)
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.
Following are the steps involved in finding the HCF of two or more monomials:
Step 1: Find the numeric coefficient of each monomial and then find their HCF.
Step 2: Find the common variables and their HCF.
Step 3: Multiply the HCF in step 1 and step 2 to get the final HCF.
2 3 4 2
4 2 3 5
3 4 2
Example 1: Find the HCF of the monomials 4x y z w , 12x y z w and 18x y z .
Solution: 4 = 2 × 2 12 = 2 × 2 × 3 18 = 2 × 3 × 3
HCF of 4, 12, 18 = 2
The common variables in the terms are x, y and z.
4
2
3
Lowest power of x, out of x , x , x = 2
3
2
4
Lowest power of y, out of y , y , y = 2
2
4
3
Lowest power of z, out of z , z , z = 2
2 2 2
HCF of the variables = x y z
2 3 4 2
4 2 3 5
3 4 2
2 2 2
So, the HCF of 4x y z w , 12x y z w and 18x y z = 2x y z