ALL Algebra Maths Formulas and theorems

Algebra Maths Formulas and theorems

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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)(a^n-1 + a^n-2b+ ........ + b^n-2a + b^n-1)

👉If n is even

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

👉If n is odd

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

👉a^-m = ¹⁄_a^m

👉¹⁄_a^-m = a^m

👉(^a⁄_b)^m = ^am⁄_a^m

👉^am⁄_aⁿ = a^m-n

👉(aⁿ)^m = a^nm

👉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 n^th Power, while n^th Power of any positive rational number is not equal to ‘a’.

Ex-   is a surd, Since there is no square 11 of any positive rational number.

           

              is not a surd, because of 9 = 3²

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

Rules of Surds:

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 + a₁x^n-1 + a₂x^n-2 + ........... + a_n-1x + a_n

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

Quadratic Polynomial:

If the maximum power of the variable used in the polynomial is 2, it is called Quadratic Polynomial.

e.g.- 2x² + 5x + 8,  x² + 5x - 10

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