Degree of Polynomial

Polynomial

We have studied algebraic expressions and polynomials. To recall an algebraic expression f(x) of the form f(x) = a0 + a1x + a2x2 + a3 x3 + ……………+ an xn, there a1, a2, a3…..an are real numbers and all the index of ‘x’ are non-negative integers is called a polynomial in x.Polynomial comes from “poly” meaning "many" and “nomial”  meaning "term" combinedly it means "many terms"A polynomial can have constants, variables and exponents.

The degree of a polynomial is nothing but the highest degree of its exponent(variable) with non-zero coefficient. Here the term degree means power. In this article let us study various degrees of polynomials.

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What is the Degree of a Polynomial?

The highest degree exponent term in a polynomial is known as its degree.

To find the degree all that you have to do is find the largest exponent in the given polynomial. 

For example, in the following equation: 

f(x) = x3 + 2x2 + 4x + 3. The degree of the equation is 3 .i.e. the highest power of the variable in the polynomial is said to be the degree of the polynomial.

f(x) = 7x2 - 3x + 12 is a polynomial of degree 2.

thus,f(x) = an xn + an-1 xn-1 + an-2xn-2 +...................+ a1 x + a0  where a0 , a1 , a2 …....an  are constants and an ≠ 0 .

On the basis of the degree of a polynomial , we have following names for the degree of polynomial.


Degree of Zero Polynomial

If all the coefficients of a polynomial are zero we get a zero degree polynomial. Any non - zero number (constant) is said to be zero degree polynomial if f(x) = a as f(x) = ax0 where a ≠ 0 .The degree of zero polynomial is undefined because f(x) = 0, g(x) = 0x , h(x) = 0x2 etc. are equal to zero polynomial.


Constant Polynomial

A polynomial having its highest degree zero is called a constant polynomial. It has no variables, only constants.

For example: f(x) = 6, g(x) = -22 , h(y) = 5/2 etc are constant polynomials. In general f(x) = c is a constant polynomial.The constant polynomial 0 or f(x) = 0 is called the zero polynomial. 


Linear Polynomials

A polynomial having its highest degree one is called a linear polynomial.

For example, f(x) = x- 12, g(x) = 12 x , h(x) = -7x + 8 are linear polynomials.

In general g(x) = ax + b , a ≠ 0 is a linear polynomial.


Quadratic Polynomial

A polynomial having its highest degree 2 is known as a quadratic polynomial.

For example, f (x) = 2x2 - 3x + 15, g(y) = 3/2 y2 - 4y + 11 are quadratic polynomials.

In general g(x) = ax2 + bx + c, a ≠ 0 is a quadratic polynomial.


Cubic Polynomial

A polynomial having its highest degree 3 is known as a Cubic polynomial.

For example, f (x) = 8x3 + 2x2 - 3x + 15, g(y) =  y3 - 4y + 11 are cubic polynomials.

In general g(x) = ax3 + bx2 + cx + d, a ≠ 0 is a quadratic polynomial.


Bi-quadratic Polynomial

A polynomial having its highest degree 4 is known as a Bi-quadratic polynomial.

For example, f (x) = 10x4 + 5x3 + 2x2 - 3x + 15, g(y) = 3y4 + 7y + 9 are quadratic polynomials.

In general g(x) = ax4 + bx2 + cx2 + dx + e, a ≠ 0 is a bi-quadratic polynomial.

Based on the degree of the polynomial the polynomial are names and expressed as follows:

Types of Polynomials Based on their Degrees 

Degree

Type

General Form

0

Constant/Zero

g(x) = c

1

Linear

g(x) = ax + b

2

Quadratic

g(x) = ax² + bx + c

3

Cubic

g(x) = ax³ + bx² + cx + d

4

Bi-Quadratic

g(x) = ax⁴ + bx³ + cx² + dx¹ +e

 

How to Find the Degree of a Polynomial?

There are simple steps to find the degree of a polynomial they are as follows:

Example: Consider the polynomial 4x5+ 8x3+ 3x5 + 3x2 + 4 + 2x + 3

Step 1: Combine all the like terms variables  

(4x5 + 3x5) + 8x3  + 3x2 + 2x + (4 + 3)

Step 2: Ignore all the coefficients and write only the variables with their powers.

           x5 + x3 + x2 + x + x0

Step 3: Arrange the variable in descending order of their powers if their not in proper order.

  x5 + x3 + x2 + x1 + x0

Step 4: Check which the  largest power of the variable  and that is the degree of the polynomial

x5 + x3 + x2 + x + x0 = 5


Solved Examples

1. What is the Degree of the Following Polynomial

i) 5x4 + 2x3 +3x + 4

Ans: degree is 4

ii)11x9 + 10x5 + 11

Ans: degree is 9

2. Find the Zeros of the Polynomial.

p(x) = 3x - 2

Solution: 

3x - 2 = 0

3x = 2

x= ⅔

x = ⅔ is a zero of p(x) = 3x - 2 


Quiz Time

1. Write the Degrees of Each of the Following Polynomials.

    1. 12-x + 2x3

    2. 4x3 + 2x2 + 3x + 7


2. Identify the Polynomial

    1. p(x) = 2x2 - x + 1

    2. h(x) = x4  + 3x3 + 2x2  + 3


Fun Facts

  • A linear polynomial has only one zero

  • If the degree of polynomial is n; the largest number of zeros it has is also n.

FAQ (Frequently Asked Questions)

1. What are Polynomials? Mention its Different Types.

Answer: Polynomial comes from the word “poly” meaning "many" and “nomial”  meaning "term" together it means "many terms"

Polynomials are algebraic expressions that may comprise of exponents, variables and constants which are added, subtracted or multiplied but not divided by a variable. Polynomials are of different types, they are monomial, binomial, and trinomial. A monomial is a polynomial having one term. A binomial is an algebraic expression with two, unlike terms. A trinomial is an algebraic expression  with three, unlike terms.

These are polynomials:

  • 3x

  • x − 2

  • 3xyz + 3xy2z

  • 512v5 + 99w5

  • Yes, "7" is also polynomial, one term is allowed, and it can be just a constant.

 These are not polynomials

  • 3xy-2 is not, because the exponent is "-2" which is a negative number.

  • 7/(x+5) is not, because dividing by a variable is not allowed

  • 1/x is not either

  • √y is not, because the exponent is "½" . 

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2. Explain Different Types of Polynomials.

Answer: Types of Polynomials

Let us get familiar with the different types of polynomials. They are as follows:

  1. Monomials –An algebraic expressions with one term is called monomial hence the name “Monomial. In other words, it is an expression that contains any count of like terms. For example, 2x + 4x + 9x is a monomial because when we add the like terms it results in 15x. Furthermore, 21x2y, 8pq etc are monomials because each of these expressions contains only one term.

  2. Binomials – An algebraic expressions with two unlike terms, is called binomial  hence the name “Bi”nomial. For example, 3x + 5x2 is binomial since it contains two unlike terms, that is, 3x and 5x2. Likewise, 12pq + 13p2q is a binomial.

  3. Trinomials – An expressions with three unlike terms, is called as trinomials hence the name “Tri”nomial. For example- 3x + 6x2 – 2x3 is a trinomial. It is due to the presence of three, unlike terms, namely, 3x, 6x2 and 2x3. Likewise, 11pq + 4x2 –10 is a trinomial.

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