Subject :- Maths

Topic :- Polynomials

Made by :- Nirav Vaishnav.

Constants : A symbol having a fixed numerical value is called a constant.Eg :- In polynomial 3x + 4y ,3 and 4 are the constants.Variables : A symbol which may be assigned different numerical values is called as a variables. Eg :- In polynomial 3x + 4y , x and y are the variables.

Algebraic Expression : The combination of constants and variables are called algebraic expressions.Eg :- 2x3–4x2+6x–3 is a polynomial in one variable x.4+7x4/5+9x5 is an expression but not a polynomial since it contains a term x4/5, where 4/5 is not a non-negative integer.

Polynomials : An algebraic expression in which the variable involved have only non –negative integral powers is called a polynomial.

Important Terms :-

Degree : The highest power of a variable in the polynomial is called degree of that polynomial.Eg. : 5x2 + 3 , here the degree is 2.

Constant polynomial : A polynomial containing one term only , consisting of a constant is called a constant polynomial.The degree of a non-zero constant polynomial is zero.Eg. : 3 , -5 , 7/8 , etc. , are all constant polynomials.

Zero polynomial : A polynomial consisting one term only , namely zero only , is called a zero polynomial.The degree of a zero polynomial is not defined.

Continued...

Types of polynomial (on the basis of terms) :-

Monomial : Algebric expression that consists only one term is called monomial.Binomial : Algebric expression that consists two terms is called binomial.Trinomial : Algebric expression that consists three terms is called trinomial.

Types of polynomial (on the basis of degree) :-

Linear polynomial: A polynomial of degree 1 is called a linear polynomial.Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial.Cubic polynomial : A polynomial of degree 3 is called a cubic polynomial.Biquadratic polynomial : A polynomial of degree 4 is called a biquadratic polynomial.

Examples :-Polynomials :- Degree :- Classified by

degree :-Classified by no. of terms :-

5 0 Constant Monomial

2x - 4 1 Linear Binomial

3x2 + x 2 Quadratic Binomial

x3 - 4x2 + 1 3 Cubic Trinomial

Remainder Theorem :-Let f(x) be a polynomial of degree n > 1 and let a be any real number.

When f(x) is divided by (x-a) , then the remainder is f(a).

PROOF :- Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder is r(x).Then, degree r(x) < degree (x-a) degree r(x) < 1 [ therefore, degree (x-a)=1] degree r(x) = 0r(x) is constant, equal to r (say)Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.Therefore, f(x) = (x-a)*g(x) + r (i)Putting x=a in (i), we get r = f(a)Thus, when f(x) is divided by (x-a), then the remainder is f(a).

Factor Theorem :-

Let p(x) be a polynomial of degree n > 1 and let a be any real number. If p(a) = 0 then (x-a) is a factor of p(x). Proof :-Let f(a) = 0On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by remainder theorem, when f(x) is divided by (x-a), then the remainder is f(a).

Therefore ,f(x) = (x-a)*g(x) + f(a)

[Since,f(a) = 0 (given)]

Therefore,(x-a) is a factor of f(x).

Algebraic Identities :-

(x+y) 2 = x 2 +2xy+y 2

(x-y) 2 = x 2 -2xy+y 2

(x+y) (x-y) = x 2 -y 2

(x+y+z) 2 = x 2 +y 2 +z 2 +2xy+2yz+2zxx 3 +y 3 = (x+y)(x 2 -xy+y 2)x 3 -y 3 = (x-y)(x 2 +xy+y 2)(x+y) 3 = x 3 +y 3 +3xy(x+y)(x-y ) 3 = x 3 -y 3 -3xy(x-y)x 3 +y 3 +z 3 -3xyz = (x+y+z)(x 2 +y 2 +z 2 -xy-yz-zx)If x+y+z =0,then x 3 +y 3 +z 3 = 3xyz

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