Factorization

In mathematics factorization is a process that is used to break down numbers into smaller numbers are discussed below.

The process of finding two or more expressions whose product is the given expression is called factorization.

Note:

Factorization is the reverse process of multiplication.

Follow the examples given below:

Product	Factorization
(i) 3x (4x - 5y) = 12x2 - 15xy	12x2 – 15xy = 3x (4x - 5y)
(ii) (x + 3)(x - 2) = x2 + x - 6	x2 + x - 6 = (x + 3)(x - 2)
(iii) (2a + 3b)(2a – 3b) = 4a2 – 9b2	4a2 – 9b2 = (2a + 3b)(2a – 3b)

Simple factorization:

Now we learn how to solve simple factorization.

1. Factorize 36x²y² – 15xy

We see that the HCF of both the terms is found. The HCF of the terms 36x²y² and 15xy is 3xy.

Each term of the given expression is multiplied and divided by the HCF.

\(3xy(\frac{36x^{2}y^{2}}{3xy} - \frac{15xy}{3xy})\)

= 3xy(12xy – 5)

2.Find the HCF of 15a³b²c³ and 12a⁴bc⁴.

The HCF of 15 and 12 = 3, common literal numbers are a, b and c. The lowest power of a = 3, b = 1 and c = 3.

Therefore, HCF of 15a³b²c³ and 12a⁴bc⁴ = 3a³bc³.


3. Find the HCF of 4a²bc, 14a³b and 2ac.

The HCF of 4, 14 and 2 = 2, common literal number is a. The lowest power of a = 1.

Therefore, HCF of 4a²bc, 14a³b and 2ac = 2a.

Remember:

(i) HCF of two or more monomials = (HCF of their numerical coefficients) × (HCF of their literal coefficients)

(ii) HCF of literal coefficients = product of each common literal raised to the lowest power.

Some solved examples:

1. Factorize 8x³ - 32x⁵

The HCF of the terms 8x³ and 32x⁵ = 8x³ (common terms in both)

Therefore, 8x³ - 32x⁵ = 8x³(1 - 4x²).


2. Factorize 14m²n⁵ - 28mn² + 7m⁵n

The HCF of the terms 14m²n⁵, 28mn² and 7m⁵n = 7mn (common terms in all)

Therefore, 14m²n⁵ - 28mn² + 7m⁵n = 7mn (2mn⁴ - 4n + m⁴).

3. Factorize 5a(b + 3c) - 5m(b + 3c)

The HCF of the terms 5a(b + 3c) and 5m(b + 3c) = 5(b + 3c)

Therefore, 5a(b + 3c) - 5m(b + 3c) = 5(b + 3c) (a - m).

Note:

Thus, factorization is the method of expressing an algebraic expression as a product of two or more expressions.

8th Grade Math Practice

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