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Polynomial: Examples, Formula, Theorem and Properties


Also, Read
Pair of Linear Equations
in Two Variables
Linear Inequalities
Quadratic Equation
Vector Algebra Formulas
Binomial Theorem


To understand about polynomials Let us first break the word poly+nomial. Where “poly” means “many” and “nomial” means “terms”. As the meaning itself suggests that it must be the mathematical expression which contains many terms. Exactly it is what is being said. It contains variables, coefficients, constants, and follows addition, subtraction and multiplication and also it contains non-negative exponents. There we listed out polynomial examples.

It is also a broader part of algebra which has its own implications in solving mathematical expressions in equations.

Terms used in polynomial:

  1. Variable: The term ‘x’ generally used in algebra is variable which takes its value according to different mathematical expression. It is also called
  2. Coefficient: The numerical which is attached to variables is known as coefficient.
  3. Constant: The numerical term in an equation which has no variable attached.

Polynomial Examples: In expression 2x+3, x is variable  and 2 is coefficient and 3 is constant term.

Notation of polynomial:

Polynomial is denoted as function of variable as it is symbolized as P(x).

Different kinds of polynomial:

There are several kinds of polynomial based on number of terms.

Monomial: The polynomial expression which contain single term.

Binomial: The polynomial expression which contain two terms.

Trinomial: The polynomial expression which contain two terms.

Polynomial Examples:
4x2y is a monomial.
2xy3 + 4y is a binomial.
4xy + 2x2 + 3 is a trinomial.

Degree of polynomial

The degree of polynomial with single variable is the highest power among all the monomials.

In terms of degree of polynomial polynomial.

  1. Quadratic Polynomial: A polynomial of degree 2 is called quadratic polynomial.
  2. Cubic polynomial: A polynomial of degree 3 is called cubic polynomial.

Zeroes of polynomial:

A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0.

Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.

Remainder Theorem:

If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial (x – a), then the remainder is p(a).
expressed as p(x) = g(x).q(x) + r(x) where, r(x) = 0 or [degree r(x)] < [degree g(x)].

Factor Theorem:

(x – a) is a factor of the polynomial p(x), if p(a) = 0. Also, if (x – a) is a factor of p(x), then p(a) = 0.

Formulae to Learn:

  • (x + y + z)2= x2 + y2 + z2 + 2xy + 2yz + 2zx
  • (x + y)3 = x3 + y3 + 3xy(x + y)
  • (x – y)3 = x3 – y3– 3xy(x – y)
  • x3 + y3 + z3 – 3xyz = (x + y + z)(x2+ y2+ z2– xy – yz – zx

Polynomial Examples:

Find the remainder when x4 + x3 – 2x2 + x + 1 is divided by x – 1.


Here, p(x) = x4 + x3 – 2x2 + x + 1, and the zero of x – 1 is 1.
So, p(1) = (1)4 + (1)3 – 2(1)2 + 1 + 1
= 2
So, by the Remainder Theorem, 2 is the remainder when x4 + x3 – 2×2 + x + 1 is divided by x – 1.

Verify whether 2 and 0 are zeroes of the polynomial x2 – 2x.


Let p(x) = x2 –  2x
Then p(2) = 22 – 4 = 4 – 4 = 0
and p(0) = 0 – 0 = 0
Hence, 2 and 0 are both zeroes of the polynomial x2 – 2x.

The following observations took place:

(i) zero of a polynomial need not be 0.

(ii) 0 may be a zero of a polynomial.

(iii) Every linear polynomial has one and only one zero.

(iv) A polynomial can have more than one zero.

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