# Algebra Maths Formulas and theorems

UPI ID:- achalup41-1@oksbi

## ALL ALGEBRA FORMULAS

👉(a + b)2 = a2 + b2 + 2ab

👉(a - b)2 = a2 + b2 - 2ab

👉(a2 - b2) = (a + b) (a - b)

👉(a + b)2 + (a - b)2 = 2(a2 + b2)

👉(a + b)2 - (a - b)2 = 4ab

👉(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

👉(a - b + c)2 = a2 + b2 + c2 + 2(-ab - bc + ca)

👉(a + b - c)2 = a2 + b2 + c2 + 2(ab - bc - ca)

👉(a + b)3 = a3 + b3 + 3ab(a + b)

👉(a - b)3 = a3 - b3 - 3ab(a - b)

👉a3 + b3 = (a + b)(a2 - ab + b2)

👉a3 - b3 = (a - b)(a2 + ab + b2)

👉a3 + b3 + c3 - 3abc = (a+b+c)(a2+b2+c2-ab-bc-ca)

If a+b+c = 0, so a3 + b3 + c3 = 3abc

👉(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

👉(a - b)4 = a4 - 4a3b + 6a2b2 - 4ab3 + b4

👉 a4 - b4 = (a - b)(a + b)(a2 + b2)

👉 a5 - b5 = (a - b)(a4 + a3b + a2b2 + ab3 + b4)

👉 a8 - b8 = (a + b)(a - b)(a4 + b4)(a2 + b2)

👉If n is a natural number

an - bn = (a - b)(an-1 + an-2b+ ........ + bn-2a + bn-1)

👉If n is even

an + bn = (a + b)(an-1 - an-2b + ......... + bn-2a - bn-1)

👉If n is odd

👉an + bn = (a - b)(an-1 - an-2b+ ........ -bn-2a + bn-1)

👉a-m1am

👉1a-m = am

👉(ab)m = amam

👉aman = am-n

👉(an)m = anm

👉a0 = 1

### Factor Theorem:

If (x-a) is the factor of f(x)

then f (a) = 0

### Remainder Theorem:

If f(x) is divided by (x-a),

then Remainder = f (a)

### Some special results

(xn - an)will always be completely divisible by ( x - a).

(xn - an)will always be completely divisible by ( x + a), While n is an even number.

(xn + an)will always be completely divisible by ( x+a), While n is an odd number.

(xn + an)will never be completely divisible by (x-a).

### SURDS

Let ‘a’ is a rational number. And ‘n’ is a positive integer. Then is called Surds of nth Power, while nth Power of any positive rational number is not equal to ‘a’.

Note: It is not necessary to have every irrational number a surd.

#### Pure Surd:

If any monolithic has coefficient 1. Then this is called Pure Surd.

#### Mixed Surd:

If a monolithic has any rational coefficient other than 1, then it’s called Mixed Surd.

### Polynomial:

p(x) = a0xn + a1xn-1 + a2xn-2 + ........... + an-1x + an

a0, a1 ........... are the positive or negative real number and n is the complete number.

e.g.- 4x3 - 2x2 + 5x - 8

#### Degree of Polynomial:

The exponent of the term with the largest degree in a polynomial is called the power of that polynomial.

e.g.- x3 + 2x2 + 5x + 8 the term with the largest degree x3, so the polynomial degree is 3.

#### Linear Polynomial:

If the maximum power of the variable used in the polynomial is 1, it is called a linear polynomial.

e.g.- 4x - 8, 5x, 6x + 4