Things to Remember: Important Formula for Algebra Math

Important Notes:

- The
**square of a number**is the number multiplied by itself. - A
**natural number***n*is a**perfect square**if*n*=*x*^{2}, for all the natural number*x*. - The
**square of every even number**is always even and the**square of every odd numbers**is always odd. - The three
**consecutive numbers**(natural numbers) let’s say*a*,*b*and*c*are called the**Pythagorean triplet**if*c*^{2}=*a*^{2}+*b*^{2}

- If we have a number
*x*, then we can easily find the**Pythagorean Triplets**by substituting the value in 2*x*, x^{2}– 1 and m^{2}+ 1. - The
**square of a natural numbers***n*is equal to the sum of the first*n*odd numbers.

For any positive numbers x and y, we have

- sqrt(ab) = sqrt a x sqrt b
- sqrt(a/b) = sqrt a / sqrt b
- sqrt(a+b) (Not equal to sign) sqrt a + sqrt b
- sqrt(a – b) (Not equal to) sqrt a – sqrt
- sqrt 2, sqrt 3, sqrt 5, . . . . . . are called the irrationals.
- The negative of an
**irrational number is irrational.** **Zero is called a real number**and it has no multiplicative inverse exists.- The number e and pi (pi: the value is 3.14 or 22/7) are
**the irrational numbers.** - Every
**Rational number**is either**rational or irrational.**

- The cube of a number is the line number raised to the power 3 (three).
- The cube of a negative integer is always negative and positive integer is always positive.
- The cube of a rational number of x/y is x
^{3}/y^{3}. - The cube of an even natural number is even and odd natural number is odd.
- If x be any positive integer, we have 3Sqrt(-x) = -3Sqrt(x).
- The cube root of a rational number x/y is given by 3Sqrt(x/y) = 3Sqrt(a)/3Sqrt(b).
- A natural number n is a perfect cube if x = y
^{2}, where y is any natural number. - The cube root of a given number is that number whose cube is equal to the given number.

Factors and Algebraic Expressions

- (x + a)(x + b) = x
^{2}+ (a + b)x + ab - a
^{2}– b^{2}= (a + b)(a – b) - (a + b)
^{2}= a^{2}+ 2ab + b^{2} - (a – b)
^{2}= a^{2}– 2ab + b^{2} - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ca) - (a + b)
^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a – b)
^{3}= a^{3}– b^{3}– 3ab(a – b) - a
^{3}+ b^{3}= (a + b)^{3}– 3ab(a + b) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2}) - a
^{3}– b^{3}= (a – b)^{3}+ 3ab(a – b) - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}+ c^{3}– 3abc = (a + b + c)(a^{2}+ b^{2}+ c^{2}-ab – bc – ca) - If a + b + c = 0, then a
^{3}+ b^{3}+ c^{3}= 3abc - If (a
^{2}+ b^{2}+ c^{2}– ab – bc – ca) = 0, then a^{3}+ b^{3}+ c^{3}= 3abc.

Factors and Algebraic Expressions

- Degree of any constant is Zero.
- The combination of constants and variables connected by +, -, x and / is called the algebraic expressions.
- A polynomial is called a monomial (having 1 term). binomial (having 2 terms) or trinomial (having 3 terms)
- Any terms with the same variable and the same exponent are called the like terms.
- Dividend is the product of Divisor and Quotient and add the Remainder.
- The degree of the remainder is always less than that of the divisor.
- An equation is the statement of equality of two algebraic expressions involving one or more unknown quantities.
- An equation involving only linear polynomials is called a linear equation.
- The equation, some of the quantity may be added or subtracted from both side of an equation without changing the equality.
- In the equation, both sides of the equation may be multiplied or divided by the same non-zero number (a number whose denominator part is not zero) without changing the equality.
- When we solve the equation, we can easily transpose a term from one side to the other side after changing its sign.
- If (ax + b)/(cx + d) = m/n then n(ax + b) = m(cx + d). This process is called the cross multiplication.