polynomials

polynomials

1. There are five types of polynomials

   • cubic polynomial
     • A polynomial of degree 3 is called cubic polynomial.
       • For example f(x)=12x^3-4x^2 +7x-6,g(x)=2x^2+4x-12 etc are cubic polynomial.
         • Thus, f(x)=ax^3+bx^3+cx^2+dx+e, a not=0 is a bi quadratic polynomial.
           • Remark 3=In above discussion we have dealt with in one variable only .We can also have one variable.For example ,x^2+y^2+2xyz(where x,y,z are variable) is a polynomial in three variable. similarly, if u,v,w, are variables, then u^2+v^8+3w is a polynomial in three variable
   • quadratic polynomial
     • A polynomial of degree 2 is known as a quadratic polynomial. for example f(x)=2x^2-3x+15,g(x)=3/2y^2-4y+11/3 etc are quadratic polynomial
       • for example f(x)=2x^2-3x+15,g(x)=3/2y^2-4y+11/3 etc are quadratic polynomia
         • In general,f(x)=as^2+bx+c,a not= 0 is a quadratic polynomial.
           • Remark 2= A quadratic polynomial may be a monomial or a binomial or a tinomial. Because, f(x)=7x^2 is a monomial, g(x)=2x^2+3 is a binomial and h(x)=3x^2-2x+4 is a trinomial
   • linear polynomial
     • a polynomial of degree one is called linear polynomial for example,f(x)=x-12,g(x)=12x,h(x)=-7x+8are linear polynomials.In general,f(x)=ax+b,a not= 0 is a linear polynomial
       • hhhhhylik
       • for example,f(x)=x-12,g(x)=12x,h(x)=-7x+8are linear polynomials.In general,f(x)=ax+b,a not= 0 is a linear polynomial
         • Remark 1= A linear polynomials may be a monomial or a binomial. Because , linear polynomial f(x)=7x-15 is a binomial where as g(x)=3x is a monomial.
   • constant polynomial
     • a polynomial with degree zero is called a constant polynomial
       • for example:f(x)=2,g(x)=-12,h(y)=3/2 etc are the constant polynomials.
         • a constant polynomial is called zero polynomialthe degree of a zero polynomial is not defined, because f(x)=0,g(x)=0 etc, are equal to zero polynomial.
   • bi-quadratic polynomial
     • A fourth degree polynomial is called a bi-quadratic polynomial.
       • For example,f(x)=10x^4-7x^3+8x^2-12x-20 is a bi-quadratic polynomial
         • In general,f(x)=ax^4+bx^3+cx2+dx+e,a not=0 is a bi-quadratic polynomial.

2. POLY(MANY)

3. Roots of polynomial

   • If x=a is such that p(a)=0 then x=a is said to be root of polynomial equation p(x)=0 (x-a) will be a factor of polynomial
     • Key Points
       • 1.Any polynomial will have maxium 1 root and it is taken as a
       • 3.Any cubic polynomial will have maxium 3 root and are taken as a,b,c
       • 2.Any quadratic polynomial will have maxium 2 root and are taken as a,b

4. NOMIAL(TERM)

5. POLYNOMIALS ARE ALGEBRIC EXPRESSIONS MADE UP OF ONE OR MORE TERM OF A PRACTICAL TYPE

6. ZERO OF A POLYNOMIAL

   • A real number (a) is a zero of a polynomial p(x) if p(a)=0

7. Standard Form

   • Coefficients
     • a0,a1,a2,an are real numbers a0 not=0 n is the whole number
       • Leading coefficient =a0
   • The terms with highest degree first then at the last constant term

8. Factorisation of polynomial

   • Algorithm
     • Algorithm to factorise a quadratic or cubic polynomial
   • step1
     • obtain the constant term x^2 unit
   • step 2
     • step 3
       • Similarly , obtain the other factors or in case of quadratic polynomial. divide the given quadratic polynomial by (x-a) to get the other linear factor.In case of cubic polynomial divide the given polynomial (x-a) to get the other factor(quadratic).Factorise the quadratic factor so obtain by the appropriate method to get the 2 linear factor.
     • Take one of the factors of the term say aand replace x by it in the given example.If the polynomial reduces to zero then (x-a) is a factor of polynomial

9. Factor Theorem

   • If p(x) is a polynomial of degree n>1 and a real number , then
     • 1. x-a is a factor of p(x) if p(a)=0 and 2. p(a)=0 if x-a is a factor of p(x)