Polynomial

Polynomial Meaning

What is a Polynomial?

  • In Mathematics, a polynomial is an expression consisting of coefficients and variables which are also known as indeterminates.

  • As the name suggests poly means many and nomial means terms, hence a polynomial means many terms.

  • Polynomials are generally a sum or difference of variables and exponents.

  • Each part of the polynomial is known as “term”.

  • Polynomial examples \[--{\mathbf{4}}{{\mathbf{x}}^{\mathbf{2}}} + {\mathbf{3x}} - {\mathbf{7}}\]


Rules for an Expression to be a Polynomial-

An algebraic expression should not consist of – 

  • Square root of variables.

  • Fractional powers on the variables.

  • Negative powers on the variables.

  • Variables in the denominators of any fractions.

  • Not a Polynomial examples,

6x-2 is not a polynomial since there is negative power on the variable.

   

√x is not a polynomial, because the variable inside is a radical



1/x2 is not a polynomial, because the variable is in the denominator.


Terminologies in Polynomial Definition-

According to polynomial definition,

\[{\mathbf{4}}{{\mathbf{x}}^{\mathbf{2}}} + {\mathbf{3x}} - {\mathbf{7}}\]

In the polynomial given above, 

\[{\mathbf{4}}{{\mathbf{x}}^{\mathbf{2}}}\]= Leading term

7 = Constant

And \[{\mathbf{4}}{{\mathbf{x}}^{\mathbf{2}}},\] 3x, and 7 are terms of the polynomial.


Now you Might Think What A Constant Is?

  • A constant is a part of the polynomial which does not consist of any variable.

  • As it does not consist of any variables, therefore its value never changes and it is known as a constant term.

  • For example \[--{\text{ }}9x{\text{ }} + {\text{ }}4\], where 4 is the constant as it does not contain any variable.


Degree of a Polynomial

The degree of a polynomial can be defined as the highest degree of a monomial within a polynomial. The degree of the polynomial can be defined as a polynomial equation having a single variable that has the greatest exponent.

Type of Polynomial

Degree of Polynomial

Example

Constants or Zero Polynomial

0

4

Linear Polynomial

1

2x+1

Quadratic Polynomial

2

\[5{x^2} + 2x + 1\]

Cubic Polynomial

3

\[5{x^3} + 4{x^3} + 2x + 1\]

Quartic Polynomial

4

\[8{x^4} + 3{x^3} + 3{x^2} + 2x + 1\]

 

Example: What is the degree of the given polynomial \[5{x^3} + 4{x^3} + 2x + 1\]

Solution: The degree of the given polynomial is 4.

Terms of a polynomial according to the polynomial definition-

The terms in a polynomial are the parts of the algebraic expression which are separated by “+” or “-” signs. Each part of the polynomial is known as “term”. For example, let us take a polynomial, say, \[4{x^2}\; + {\text{ }}5\]

In the polynomial given below the number of terms is 2. We can classify a polynomial on the basis of the number of terms in the polynomial.

                      

Types of Polynomial –

There are three types of polynomials.

  1. Monomial

  2. Binomial 

  3. Trinomial 


What is a Monomial?

An algebraic expression that consists of only one term is known as a monomial. If the single term is a non-zero term then only the algebraic expression is known as a monomial. Here, are a few examples of monomials:

  • 6x

  • 2

  • \[\;2{a^4}\]


What is a Binomial?

An algebraic expression that consists of two terms is known as binomial. A binomial is generally represented as a sum or difference of two or more monomials.  Here are a few examples of binomials:

  • – 9x+2,

  • \[6{a^4}\; + {\text{ }}11x\]

  • \[x{y^2} + 4xy\]


What is a Trinomial?

An algebraic expression which consists of three terms is known as trinomial Here are a few examples of trinomial expressions:

  • \[4{a^4} + 3x + 2\]

  • \[7{x^2}\; + {\text{ }}8x{\text{ }} + {\text{ }}5\]

  • \[8{x^3} + 2{x^2} + 7\]

 

Degree of polynomial- 

The highest or the greatest degree of a variable in any polynomial is known as the degree of the polynomial. The highest exponential power in the polynomial is the degree of the polynomial.

What do you mean by the degree of a Zero Polynomial?

If in a given polynomial all the coefficients are equal to zero, then the degree of the zero polynomial is either set equal to -1 or is undefined.

What do you mean by Constant Polynomials?

A constant polynomial contains no variables and the value of the polynomial does not change as there are no variables. In a constant polynomial, the power of the variable is equal to zero. Any constant can be expressed with a variable with its exponential power equal to zero. Constant term = 4, Polynomial form P(x)= 4x0

Solving Polynomials-

We can solve any polynomial using factorization and basic concepts of algebra. The first step to solve a polynomial is to set the right-hand side of the polynomial as 0.

There are two types of polynomials you need to know about!

  1. Solving Linear Polynomials

  2. Solving Quadratic Polynomials

 

How to Solve Linear Polynomials?

It is very easy to solve linear polynomials! 

Step 1) Equate the given equation with 0, Make the equation equal to zero.

Step 2) Then solve the equation, using basic concepts of algebra.

How to Solve Linear Polynomials?

To solve a quadratic polynomial, 

Step 1) We need to rewrite the given expression in the ascending order of degree. 

Step 2) Equate the equation with zero.

Step 3) Use the factorization method to solve the equation.

Questions to be solved-

Question 1) What is the degree of the polynomial \[9{x^4}\; + {\text{ }}4{x^3} + {\text{ }}1\]

Solution) The degree of the given polynomial \[9{x^4}\; + {\text{ }}4{x^3} + {\text{ }}1\]is 4 since the highest degree in the given polynomial is 4.

Question 2) What is the degree of the polynomial \[6{x^8} + {\text{ }}9{x^3}\]?

Solution) 8 is the degree of the polynomial \[6{x^8} + {\text{ }}9{x^3}\], since the highest degree in the given polynomial is 8.

Question 3) Solve the given polynomial 4x – 8.

Solution) Let’s put zero on the right-hand side,

4x – 8 = 0

⇒ 4x = 8

⇒ x = 8/4

x= 2

Thus, the value of x= 2

FAQ (Frequently Asked Questions)

1) What is a Polynomial and what is not a Polynomial?

If an algebraic expression consists of a radical then it is not a polynomial. In order to a polynomial, all the exponents present in the algebraic expression should be non-negative integers.

2. What is a Polynomial with four terms known as?

A polynomial with four terms is generally known as quadrinomial. Since the number of terms is not important in a polynomial, therefore there is no need for such terms.

3. What is not a Polynomial?

If an algebraic expression consists of square roots of variables, fractional or negative powers on the variables and variables in the denominators of any fractions, then it is not a polynomial.

 Not a polynomial example, 6x-2 is not a polynomial since there is negative power on the variable. √x is not a polynomial, because the variable inside is a radical.