Degree of a polynomial 

You might have heard about polynomials in your math classes. A polynomial is just an expression consisting of variables and coefficients. It involves addition, subtraction, multiplication, and non-negative integer exponentiation of variables. The degree of a polynomial is the highest exponent of the variable in a polynomial expression. Ex: f(x) = 2x2 + 7x + 6. Here the degree of the polynomial is 2. 

Arithmetic operations like addition, subtraction, and multiplication can be performed on polynomials. But the division of a polynomial can be performed by synthetic division method or by performing long division. Let us now discuss the degree of the polynomial and the synthetic division method in detail.

How to Find the Degree of a Polynomial?

Consider a polynomial expression: f(x): x5−12x4+ 3x2 − 8. The term with the highest exponent of x is x5 and the corresponding power is 5. Therefore, the degree of this polynomial is 5. Thus, in a single variable polynomial, the degree of a polynomial is the highest exponent of the variable. But, in a polynomial of multiple variables, the degree of the polynomial can be obtained by adding the powers of different variables in any terms present in the polynomial expression.  

Example: 3×3 + 6x2y2 + 3y3+5 In this expression the degree of the polynomial is found by adding the exponents of the variables. Here we add the exponents of x2y2 = 2+2 = 4. So 4 is the degree of the given polynomial. 

Degree of a Zero polynomial

When all the coefficients in a polynomial are equal to zero; then it is called a zero polynomial. Hence, the degree of the zero polynomial is either undefined or defined in a negative way (-1 or ∞).

Degree of a Constant Polynomial

A constant polynomial has no variables.Ex: f(x) = c. Since the variables are absent, there won’t be any exponents. Thus, zero is the degree of the constant polynomial. Example: For f(x) = 8 or 8x0 the degree of the polynomial = 0. 

Mathematical Operations on Polynomials

• Addition of polynomials: Let x2 +3x+5 and 3x2+6x+9 be the two polynomials.

We have to add the variables of same exponents to perform addition

 x2 + 3x + 5 + 3x2+ 6x + 9

= 4x2 + 9x + 14

• Subtraction of polynomials: Let 3x2+6x+9 and x2+3x+5 be the two polynomials. Then the subtraction is carried out as shown below

3x2+6x+9 – (x2+3x+5) = 3x2 – x2 + 6x – 3x + 9 – 5

=2x2 + 3x + 4

• Multiplication of polynomials: Polynomials can be multiplied as shown below.

Example: (3x – 2y)(4x + 5y) 

= 3x(4x + 5y) – 2y(4x + 5y) 

= 12x2 + 15xy – 8xy – 10y2 

= 12x2 + 7xy – 10y2.

• Division of polynomials: division of a polynomial can be performed in two ways.

a.By using synthetic division method b. Long division method

1. Synthetic division method: It is a shortcut method to find the zeros of polynomials. In the synthetic division method, we are dividing a polynomial by a 1-degree polynomial. Example: Divide 2x2 – 5x + 6 by (x – 2)

Solution: The synthetic division is carried in this way

Here the coefficients are taken, then it is divided by 2. The number 2 is obtained by equating x-2 = 0 hence x = 2.

The first term is taken down as it is then it is multiplied by 2. We got 4, now write this below the second term. By adding -5 with 4 we got -1. Now repeat the steps again, that is multiplying 2 with -1, we got -2. Write it below the next term to carry out subtraction, after subtracting it with 6 we got 4. Thus the quotient obtained after synthetic division is 2x – 1 and the remainder are 4. The long division is performed similarly as the normal division. For more examples on synthetic division log on to the Cuemath website.