Algebra - Theory of equation

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Theory of equation 

If  a_0, a_1, a_{2…………..}an^-1 a_n  are integers then an expression of the form

    a₀ xⁿ+a₁x^n-1+……….+a_n . called a polynomial of degree 

n ≥  in the              variable “x”,   if a≠ 0.

Eg: -6x+y=0 , x²+x+1=0 , 6x²+3x+2=0

A polynomial of degree one is of the form ax+b=c a polynomial of degree two is of the form ax²+bx+c=0 and so on. 

Equality of 2 polynomials

Two polynomial f(x) and g(x) are said to equal if they are of same degree and the co-efficient corresponding powers are of  “x”  are equal.

Eg: f(x)=a₀ xⁿ+a₁ x^n-1+….+a_n

       g(x)=b₀y^m+b₀y^m-1+….+b_m

the value of a polynomial if f(x) is equal to ax₀+a₁ x+…+a_n is a polynomial of degree n then f(x)=a₀α+α₁x^n-1+….+α_n is called the value of the polynomial f(x) at  x= α .

·       if   f(x)=2x-7x+8

                 f(0)=8

 Zeros of polynomial or a roots of a polynomial equation. Let f(x) be a polynomial of degree m . If f(x)=0 for a real number α, then α is called a zero of the polynomial f(x) or root of a polynomial f(x).

Eg: f(x)=x²-5x+6

      f(2)=(2)²-5(2)+6=4-10+6

      f(3)=(3)²-5(3)+6=9-15+6

·       ·       therefore , roots are 2 and 3.                                                                                                              

                      Euclid’s algorithm or  Division Algorithm

If f(x) and g(x) are any 2 polynomial in x. Then there exist two polynomial q(x) and r(x) such that f(x)=q(x)g(x)+r(x) where either r(x)=0

or degree r(x) will be less than degree of g(x).

Eg: f(x)=x⁵-3x⁴+x³-3

      g(x)=x²+x-1

     f(x)=3x²-2x+1

     r(x)=x⁵+x-2

                                                                                                                     Mysore city- India, 2014