## Tuesday, February 11, 2020

### -:Real numbers:-

#### Euclid's Division Lemma:-

It's state that, for any two positive (let a and b), there exist two unique integers (let q and r), such that
a= bq + r, where 0≤r<b.
Here, a= Dividend ,b= Divisor ,
q= Quotient and r= Remainder

### -:Polynomials:-

#### Relationship between zeroes & coefficients of a Polynomial:

Let α and β be the zeroes of the quadratic polynomial p(x)= ax²+bx+c, a≠0, then (x-α) and (x-β) both are factors of p(x).
Therefore, ax² + bx +c = k (x-α) (x-β)
= k[x² -(α+β)x+αβ]
Where,k is some constant.

On comparing the coefficients of x²,x and constant term from both sides, we get a=k, b=-k(α+β)
and c=kαβ

#### Formation of quadratic and cubic polynomial

• If α and β are the zeroes of quadratic polynomial then quadratic polynomial=x²-(sum of zeroes)x+ product of zeroes,i.e. x²-(α+β)x+αβ
• If α,β and γ are the zeroes of the cubic polynomial, then cubic polynomial= x³-(sum of zeroes)x²+(sum of the product of zeroes taking two at a time)x-product of zeroes   i.e.x³-(α+β+γ)x²+(αβ+βγ+γα)x-αβγ

### Algebraic Methods for Solving a pair of Linear Equations:-

Some of the algebraic methods for solving a pair of linear equations are:
1. Substitution method
2. Elimination method
3. Cross-Multiplication method

#### Substitution method

In this method, valu of one variable can be found out in the terms of other variable from one of the given equation and this value is substituted in other equation, then we get an equation in one variable, which can be solved easily. For example:-

#### Elimination method

In this method, one variable out of the two variable equal in both the equations.
After eliminating that variable, the left equation is an equation in one variable, which can be solved easily. For example:
Remembering technique:-

This is also called as Shridharacharya's rule.

Nature of the roots of a quadratic equation

### - :Arithmetic Progression:-

To Find the general term of an AP
The nth term of an AP is called its general term.

To Find the nth term from the end of an AP

Arithmetic Mean
Arithmetic Mean between a and b is ½(a+b)
To Find Sum of n terms of an AP