Fundamentals of Polynomials

Fundamentals of Polynomials
Polynomial Definition
Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. An algebraic expression p(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + … aⁿ is a polynomial where a₀, a₁, ………. a_n are real numbers and n is non-negative integer
Standard Form of a Polynomial
\(P\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + .... + {a_1}x + {a_0}\)
Where a_n, a_n-1, a_n-2, ……………………, a₁, a₀ are called coefficients of x_n, x_n-1, x_n-2, ….., x and constant term respectively and it should belong to real number (\(\in\)R)
The general form to represent the polynomial is as follows:
Notation
The polynomial function is denoted by P(x) where x represents the variable.
For example, P(x) = x²-5x+11
If the variable is denoted by a, then the function will be P(a)
Polynomial in one variable :- An Algebraic expression of the form  , 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 polynomial :- The degree of the polynomial 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:-\(7{x^3}{y^3} + 4{x^2}{y^5} - 1\)  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 \(7{x^2} - 10{x^3}{y^3}{z^2} + {x^2}.{y^3}\) 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:-\(5 = 5 \times 1 = 5 \times {x^0}\)

Note 4: If P(x) = a₀ + a₁x + a₂x² + …… + a_nxⁿ is a polynomial such that deg(P) = n = 0 then, P has at most “n” distinct roots.