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

**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.

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