Degree of polynomial: Definition, Types, Graph & Sample Questions

The degree of Polynomials becomes an important concept when it comes to the application of counting the number of times the function will cross the x-axis when the function is plotted on the graph. This gives us the solution to the polynomials.  A polynomial is defined as an expression of more than 2 algebraic terms that contain different powers of the same or different variable.

Read Also: Types of Polynomials 

What is the degree of a polynomial?

The degree of a polynomial is the highest power of a variable in a polynomial equation.

One has to keep in mind that while finding the degree of a polynomial, one has to consider only the variable and ignore the coefficients.

For example,

x² +4x +3 is a polynomial.

Where x² is a leading term, the coefficients of a polynomial are 1 and 4. The degree of a polynomial is 2

If a polynomial has more than one variable, then the power of both the degrees is added and then the degree is calculated.

For example,

x⁵y³z + 2xy³ + 4x²yz²: here the degree of the polynomial is 9 because the summation of the degrees of both the variables of the first term is highest.

The video below explains this:

Polynomials Detailed Video Explanation:

Types of polynomials

Based on their degrees:

• Degree of a zero polynomial: A zero polynomial is one where all the coefficients are equal to zero. So the degree is either undefined or is set to -1. For example, 0x²

• Degree of a constant polynomial: It is a polynomial whose value remains the same. It contains no variables, thus the power of such polynomials is 0. For example, 6x⁰

• Linear polynomial: A polynomial having its highest degree 1. For example, 12x

• Quadratic polynomial: A polynomial having its highest degree as 2. For example, 2x²-3

• Cubic polynomial: A polynomial having the degree 3 is known as a Cubic polynomial. For example, 8x³+14

• Bi-quadratic polynomial: A polynomial having the degree 4 is known as a Bi-quadratic polynomial. For example, 3x⁴+x+7

How to find the degree of a polynomial?

Following are the steps to find the degree of a polynomial:

Consider the polynomial:4x⁶+8x³+3x⁵+3x²+4+2x+3

• Step 1: combine all the like terms.

(4x⁶+3x³) + 8x³ +3x²+4+2x+3

• Step-2: Ignore all the coefficients x⁹+x²+x+x⁰

• Step-3: Arrange the variable in descending order in order of their powers

x⁹+x²+x+x⁰

• Step-4: Check the largest power and that is the degree of variable x⁹+x²+x+x⁰=9

Sample questions

1. 5x⁴+2x³+3x+4

Degree is 4

2. 11x⁹+10x⁵+11

Degree is 9

3. 12-2x

Degree is 1

4. 5x-5x³

Degree is 3

5. 6x⁸-5xy

Degree is 8

Degrees Of Polynomial: Applications

• To find the number of times, the function will cross the x-axis
• To find the number of maximum solutions, that function can have
• To check whether the function is homogeneous or not.

Read More : Important Notes on Polynomials

Importance Of Degree Of Polynomials:

• The concept of the degree of the polynomial can also be applied to the degree of equations.F9x)=0 is called the degree of the equation where f(x) is a polynomial.
• We can understand the shape of a polynomial on a graph using the degree of a polynomial.
• The number of zeros of a polynomial is equal to a degree of a [polynomial.

Graphs for different types of polynomials:

1. Quadratic equation: 2x²-8x+6

Shape: parabolic

2. Linear polynomial: 2x+6

Shape: straight line

3. Cubic equation: 3x³+10x²+4x-8

Ques: Write the degree of each of the following polynomials:

Ans. 

• a) 5x³+4x²+7x

Here the highest degree is 3 so the degree of a polynomial is 3.

• b) 4-y².

Here the highest degree of a polynomial is 2 so the degree of a polynomial is 2.

• c) 5t-71/2

Here the highest exponent is 1, so the degree of a polynomial is 1.

• d) 3

As 3 can be written as 3x⁰, so the degree of a polynomial is 0

Ques: Classify the following as linear, quadratic, and cubic polynomials:

Ans. 

1. X²+X: the degree of a polynomial is 2 so it is a quadratic polynomial
2. X-x³: the degree of a polynomial is 3, so it's a cubic polynomial
3. Y+y²+4: the degree is 2 so it is a quadratic polynomial
4. 1+x: the degree is 1 so it is a linear polynomial
5. 3t: the degree of a polynomial is 1 so it is a linear polynomial
6. R²: the degree of a polynomial is 2 so it is a quadratic polynomial

7x³: the degree of a polynomial is 3 so it is a cubic polynomial.