POLYNOMIALS

Introduction : A literal symbol which can take various numerical values is called a variable (or) literal.
Ex:- a, b, c, x, y, z ...........etc
                                   \( a,b,c,x,y,z \in R \)
Constant : A quantity which remains always the same is called constant.
    Ex:-  \( 5,\,\,\sqrt 7 ,\,\,\frac{{ - 3}} {2} \)etc
Generally constants are always rerpresented by k (or) c.
Note:- i) In 2x - 7 is a constant and ‘x’ is a variable.
     ii) In 3x + k, x is any real number

Term :- Numericals or literals or their combinations by operation of multiplication are called terms.
    Ex:-\( 5,x, - 7x^2 \) etc
Constant terms :- A term of a expression having no literal is called a constant term.
    Ex:- \( - \sqrt {\frac{7} {2}} ,\frac{{\sqrt {13} }} {5} \), etc
Algebraic Expression :- A combination of terms formed by the operations like Addition and subtraction is called an Algebraic Expression.
    Ex:- \( \frac{3} {2}a + \frac{7} {5}b,\,\,8p - \sqrt 5 q + \frac{{11}} {5} \) etc
    Like terms and unlike terms:- The terms having the same literal factors are called like terms and  these having different literal  factors are called unlilke terms.
Ex:- \( 3a^2 b,\frac{{ - 5}} {2}ba^2 \) are like terms
      \( 7ab^2 ,\frac{1} {2}a^2 b \)are unlike terms
    
Types Algebraic Expressions:-
i) Monomials :- Expressions with single term are called monomials
    Ex:- 4x², 7, -8abc
ii) Binomials:- Expression with two terms are called Binomials.
    Ex:- 7x+4, 2a²b - 8c, \( x + \frac{1} {x} \)
iii) Trinomials :- Expression with three terms are called trinomials.
iv)  Multinomials :- Expressions with two (or) more terms are called multinomials.
    Ex:- \( a \pm b,\,\,2x \pm 3y \pm z \)

Polynomials:-
    Polynomial in one variable :- An Algebraic expression of the form \( a + bx + cx^2 + dx^3 \), where a, b, c and d are called constants and ‘x’ is a variable is called a polynomial.  The powers of the variable involved are non- negative Integers.

Degree of a poynomial :- The degree of the p[olynomial is the greatest power of the variable present in the polynomial.
    Ex:- 1) 7 + 5x is a polynomial of degree 1
            2) 5x³+3x²+2x-8 is a polynomial of degree 3

Polynomial in two or more variable :- 
    It is an Algebraic expression involving two or more variables with non-negative integral powers.
    Ex:- \( 7x^3 y^3 + 4x^2 y^5 - 1 \) is a polynomial in ‘2’ variables x and y.
Note 1:- In such a polynomial degree of any term is the greatest sum of the powers of the variables.
    Ex:- Degree of  \( 7x^2 - 10x^3 y^3 z^2 + x^2 .y^3 \) is 8 (3+3+2=8)
Note 2: Terms where the powers of the variable are negative or fractional i.e.  \( x^{ - 2} \) (or) \( \frac{1} {{x^2 }},\frac{1} {{y^2 }},\frac{z} {{x^2 }},\frac{{x^4 }} {y},x^{ - \frac{1} {2}} \) etc don’t form a polynomial.
Note 3:- Every non- zero number is considered a monomial with degree zero.
    Ex:- \( 5 = 5 \times 1 = 5 \times x^0 \)

Zero Polynomial : If all the coefficients in a polynomial are zeroes, then it is called a polynomial
\( O = 0.x^n + 0.x^{n - 1} + 0.x^{n - 2} + ................ \)
clearly all coefficients are zeroes and also degree is not defined as we can’t say value of ‘n’.

Zero of the polynomial :- The number for which the value of a polynomial is zero is called zero of the polynomial.

Ex- let 3x - 1 be a polynomial by putting \( x = \frac{1} {3} \)
             \( 3 \times \frac{1} {3} - 1 = 1 - 1 = 0 \)
     \( \therefore \) for \( x = \frac{1} {3} \), the polynomial 3x - 1 becomes zero
     \( \therefore \) is called zero of the polynomial
Substitutions: The method of replacing numerical values in the place of literal numbers is called  substitution.
    EX:- The value 7z at z = -1 is 7(-1) = -7