# Factorization

The **factor** is a number or algebraic expression that divides another number or expression evenly i.e its remainder is 0. (or) factors are small numbers when multiplied gives other numbers.

**For Example** 1, 2, 4, 7, 14, 28 are factors of number 28.

**Prime factor form:**If we write a number in form of the product of prime factors. Then it is called prime factors form.

Example:70 = 2 * 5 * 7; (here 2, 5, 7 are factors of 70; specially these are also called prime factors as these (2, 5, 7 are prime numbers)

**Algebraic expression factors:**Similarly, we can express algebraic expressions as the product of their factors. If an algebraic expression cannot be reduced further then it is its factor.

Example:8xy = 8 * x * y (here 8xy is formed by multiplication of numbers(8, x, y) are factors of that number)

**Factorization**

Factorization is nothing but writing a number as the product of smaller numbers. It is the decomposition of a number (or) mathematical objects into smaller or simpler numbers/objects. The process includes in factorization are:

**1. Method of Common Factors**

**Step 1:**First, split every term of algebraic expression into irreducible factors**Step 2:**Then find the common terms among them.**Step 3:**Now the product of common terms and the remaining terms give the required factor form.

**Example: Factorise 3x + 18?**

**Solution:**

Step 1:First splitting every term into irreducible factors.3x = 3 * x;

18 = 2 * 3 * 3;

Step 2:Next step to find the common term3 is the only common term

Step 3:Now the product of common terms and remaining terms is 3(x + 6)So 3(x + 6) is the required form.

**2. Factorization by Regrouping**

Sometimes the terms of the given expression should be arranged in suitable groups in such a way. So that all the groups have a common factor.

**Example 1: Factorise x ^{2 }+ yz + xy + xz?**

**Solution:**

Here we don’t have a common term for all. So we are taking (x

^{2 }+ xy) as one group and (yz + xz) as another group.Factor form of (x

^{2}+ xy) = (x * x) + (x * y)= x(x + y)

Factor form of (yz + xz) = (y * z) + (x * z)

= z(x + y)

After combining them,

x

^{2}+ yz + xy + xz = x(x + y) + z(x + y)Taking (x + y) as common we get,

x

^{2}+ yz + xy + xz = (x + y) (x + z)

**Example 2: Factorise 2xy + 3 + 2y + 3x?**

**Solution:**

2xy + 2y + 3x + 3 [here we are

rearranging termsto check if we get common terms or not]2y (x + 1) + 3(x + 1)

(2y + 3) (x + 1)

**3. Factorization using Identities**

There are many standard identities. Some of them are given below:

i. (a + b)^{2} = a^{2} + 2ab + b^{2}

ii. (a – b)^{2} = a^{2} – 2ab + b^{2}

iii. a^{2} – b^{2} = (a + b) (a – b)

**Example 1: Factorise x ^{2} + 8x + 16?**

**Solution:**

This is in the form of (a + b)

^{2}= a^{2}+ 2ab + b^{2}x

^{2}+ 8x + 16 = x^{2}+ 2 * x * 4 + 4^{2}= (x + 4)

^{2}

^{ }=^{ }(x + 4) (x + 4)

**Example 2: Factorise a ^{2} – 20a + 100?**

**Solution:**

This in the form of (a – b)

^{2}= a^{2}– 2ab + b^{2}a

^{2}– 20a + 100 = a^{2}– 2 * a * 10 + 10^{2}= (a – 10)

^{2}= (a – 10) (a – 10)

**Example 3: Factorise 25x ^{2} – 49?**

**Solution:**

This in the form of a

^{2}– b^{2}= (a + b) (a – b)25x

^{2}– 49 = (5x)^{2}– 7^{2}= (5x + 7) (5x – 7)

**4. Factors of the form (x + a)(x + b)**

In this method we need to factorise given expression such that (x + a) (x + b) = x^{2} + (a + b)x + ab.

**Example: Factorise m ^{2} + 10m + 21?**

**Solution:**

This is in the form of (x + a) (x + b) = x

^{2}+ (a + b)x + abWhere x = m; (a + b) = 10; ab = 21;

On solving we get a = 3; b = 7;

On substitution we get

m

^{2}+ 10m + 21 = m^{2}+ (3 + 7)m + 3 * 7= (m + 3)(m + 7)

**Division Of Algebraic Expressions**

**1. Division of monomial by a monomial**

**Example 1: Divide 35abc by 5ab?**

**Solution:**

Convert each term into irreducible form

35abc = 5 * 7 * a * b * c

5ab = 5 * a * b

Normal division,

35abc / 5ab = 5 * 7 * a * b * c / 5 * a * b

= 7c

**Example 2: Divide 14x ^{5} by 2x^{3}?**

**Solution:**

14x

^{5}= 2 * 7 * x * x * x * x * x2x

^{3}= 2 * x * x * x14x

^{5}/ 2x^{3}= 2 * 7 * x * x * x * x * x / 2 * x * x * x= 7x

^{2}

**2. Division of polynomial by a monomial**

**Example: Divide 8x ^{4} – 16x^{3} + 12x^{2} + 4x by 4x?**

**Solution: **

8x

^{4}– 16x^{3}+ 12x^{2}+ 4x / 4x=> 4x(2x

^{3}– 4x^{2}– 3x – 1) / 4x (Taking 4x as common and dividing it with denominator)=> 2x

^{3}– 4x^{2}– 3x – 1

**3. Division of polynomial by a polynomial**

**Example: Divide 16a ^{2} + 8 by 4a + 2?**

**Solution: **

16a

^{2}+ 8 / 4a + 2=> 4a(4a + 2) / (4a + 2) (here we took 4a common)

=> 4a