What is the maximum number of zeroes that a linear polynomial has/have:

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 “nominal”  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 a non-zero coefficient. Here the term degree means power. In this article let us study various degrees of polynomials.

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 .

Based on the degree of a polynomial, we have the following names for the degree of the 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 Aries 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 

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 + 33 + 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 = 2/3 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

What is the Importance of the Degree of a Polynomial?

The degree of a polynomial is important. It helps in finding whether the given polynomial expression is homogeneous or not. To find the homogeneity of a polynomial expression, you have to find the degree of each term of the polynomial. For example, 2x3 + 3xy2 + 5y3 is a multivariable polynomial. If you have to find the homogeneity of this polynomial, you have to find the degree of each term. If the degrees of all terms are equal then the polynomial is homogeneous. If the degrees of all terms are not equal, then the polynomial is not homogeneous. In the above example, the degree of all terms is 3, thus the given expression is homogeneous. 

Applications of Degree of a Polynomial

Some applications of degree of a polynomial are given here:

• It helps in finding the maximum number of solutions that a function can have.

• It helps in finding the maximum number of times a function can cross the x-axis when graphed.

• It helps in finding if the given polynomial is homogeneous or not

Important Points to Remember

Degree of a polynomial with one variable is the highest exponent value of the variable in the given polynomial.

Degree of a polynomial with more than one variable can be found by adding the exponents of each variable in the given terms, and then find which term has the highest degree. That is the degree of the polynomial.

Degree of a rational expression can be found by taking the degree of the numerator and subtracting the degree of the denominator.

The degree of a polynomial expression with a square root is taken as ½

Fun Facts

• A linear polynomial has only one zero

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