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 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?
Step 1: First splitting every term into irreducible factors.
3x = 3 * x;
18 = 2 * 3 * 3;
Step 2: Next step to find the common term
3 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 x2 + yz + xy + xz?
Here we don’t have a common term for all. So we are taking (x2 + xy) as one group and (yz + xz) as another group.
Factor form of (x2 + xy) = (x * x) + (x * y)
= x(x + y)
Factor form of (yz + xz) = (y * z) + (x * z)
= z(x + y)
After combining them,
x2 + yz + xy + xz = x(x + y) + z(x + y)
Taking (x + y) as common we get,
x2 + yz + xy + xz = (x + y) (x + z)
Example 2: Factorise 2xy + 3 + 2y + 3x?
2xy + 2y + 3x + 3 [here we are rearranging terms to 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 = a2 + 2ab + b2
ii. (a – b)2 = a2 – 2ab + b2
iii. a2 – b2 = (a + b) (a – b)
Example 1: Factorise x2 + 8x + 16?
This is in the form of (a + b)2 = a2 + 2ab + b2
x2 + 8x + 16 = x2 + 2 * x * 4 + 42
= (x + 4)2
= (x + 4) (x + 4)
Example 2: Factorise a2 – 20a + 100?
This in the form of (a – b)2 = a2 – 2ab + b2
a2 – 20a + 100 = a2 – 2 * a * 10 + 102
= (a – 10)2
= (a – 10) (a – 10)
Example 3: Factorise 25x2 – 49?
This in the form of a2 – b2 = (a + b) (a – b)
25x2 – 49 = (5x)2 – 72
= (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) = x2 + (a + b)x + ab.
Example: Factorise m2 + 10m + 21?
This is in the form of (x + a) (x + b) = x2 + (a + b)x + ab
Where x = m; (a + b) = 10; ab = 21;
On solving we get a = 3; b = 7;
On substitution we get
m2 + 10m + 21 = m2 + (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?
Convert each term into irreducible form
35abc = 5 * 7 * a * b * c
5ab = 5 * a * b
35abc / 5ab = 5 * 7 * a * b * c / 5 * a * b
Example 2: Divide 14x5 by 2x3?
14x5 = 2 * 7 * x * x * x * x * x
2x3 = 2 * x * x * x
14x5 / 2x3 = 2 * 7 * x * x * x * x * x / 2 * x * x * x
2. Division of polynomial by a monomial
Example: Divide 8x4 – 16x3 + 12x2 + 4x by 4x?
8x4 – 16x3 + 12x2 + 4x / 4x
=> 4x(2x3 – 4x2 – 3x – 1) / 4x (Taking 4x as common and dividing it with denominator)
=> 2x3 – 4x2 – 3x – 1
3. Division of polynomial by a polynomial
Example: Divide 16a2 + 8 by 4a + 2?
16a2 + 8 / 4a + 2
=> 4a(4a + 2) / (4a + 2) (here we took 4a common)