Question

A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = anx n + an−1x n−1 + ....... + a2x 2 + a1 x + a0, where a0, a1, a2, .... an are respectively the coefficients of x 0 , x 1 , x 2 , .... x n and n is called the degree of the polynomial if an ≠ 0. Each anx n ; an−1x n−1 ; .... a0, is called a term of the polynomial p(x)

Answer 1

POLYNOMIALS

Definition of a polynomial.

A polynomial $p(x)$ in one variable $x$ is an algebraic expression in $x$ of the form

$p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}$

where $a_{0},a_{1},a_{2},...,a_{n}$ are respectively the coefficients of $x^{0},x^{1},x^{2},...,x^{n}$ and $n$ is called the degree of the polynomial if $a_{n}\neq 0$.

Each $a_{n}x^{n}$; $a_{n-1}x^{n-1}$; ...; $a_{0}$ is called a term of the polynomial $p(x)$.

Condition of equality of two polynomials.

The two polynomials of the same degree

• $p(x)=a_{0}x^{n}+a_{1}x^{n-1}+...+a_{n}$

• $q(x)=b_{0}x^{n}+b_{1}x^{n-1}+...+b_{n}$

is said to be equal or identical if

• $a_{0}=b_{0},a_{1}=b_{1},...,a_{n}=b_{n}$

Examples of a few polynomials.

Some polynomials are given:

• $f(x)=x^{4}-4x^{3}+3x^{2}+3x+7$

• $f(x)=x^{5}+5x^{4}+x^{3}+5x^{2}+2x+11$

• $f(x)=x^{6}+x^{3}+1$

• $f(x)=x^{10}+x^{7}+x^{4}+x^{3}+1$

Answer 2

What is Polynomial

• A function p(x) of the form p(x) = a₀ + a₁x + a₂x² + a₃x³ + ........ + aₙxⁿ is called a Polynomial

• Expression mixed (or composed) of variable, constants and exponents is known as Polynomial

What is Degree of Polynomial

• Degree of Polynomial is termed as the Highest exponent of the variable in a polynomial is known as a Degree of a polynomial.

How To Find Degree of Polynomial (Single Variable)

Degree of a Polynomial with a single variable is it's Highest Power

Example : 6x⁴ + 3x³ +3x² + 2x + 1

• Variable in this polynomial is "x"
• The Highest Power of "x" is 4
• Hence Degree of Given Polynomial is 4

How To Find Degree of Polynomial (Two or More Variables)

Degree of a Polynomial with more than single variable is the sum of the powers of the variable in each term and the Highest sum among them is considered as Degree of that Polynomial

Example : 4x²y - 2xy² + x - 7

• Variables in this polynomial are "x" and "y"

• The Power of Term 4x²y is 3

• The Power of Term 2xy² is 3

• The Power of Term x is 1

• The Power of Term 7 is 0

• Hence Degree of Given Polynomial is 3