Polynomial Operations

There are four main polynomial operations which are:

• Addition of Polynomials
• Subtraction of Polynomials
• Multiplication of Polynomials
• Division of Polynomials

Each of the operations on polynomials is explained below using solved examples.

Addition of Polynomials

To add polynomials, always add the like terms, i.e. the terms having the same variable and power. The addition of polynomials always results in a polynomial of the same degree. For example,

Example: Find the sum of two polynomials: 5x³+3x²y+4xy−6y², 3x²+7x²y−2xy+4xy²−5

Solution:

First, combine the like terms while leaving the unlike terms as they are. Hence,

(5x³+3x²y+4xy−6y²)+(3x²+7x²y−2xy+4xy²−5)

= 5x³+3x²+(3+7)x²y+(4−2)xy+4xy²−6y²−5

= 5x³+3x²+10x²y+2xy+4xy²−6y²−5

Subtraction of Polynomials

Subtracting polynomials is similar to addition, the only difference being the type of operation. So, subtract the like terms to obtain the solution. It should be noted that subtraction of polynomials also results in a polynomial of the same degree.

Example: Find the difference of two polynomials: 5x³+3x²y+4xy−6y², 3x²+7x²y−2xy+4xy²−5

Solution:

First, combine the like terms while leaving the unlike terms as they are. Hence,

(5x³+3x²y+4xy−6y²)-(3x²+7x²y−2xy+4xy²−5)

= 5x³-3x²+(3-7)x²y+(4+2)xy-4xy²−6y²+5

= 5x³-3x²-4x²y+6xy-4xy²−6y²+5

Multiplication of Polynomials

Two or more polynomial when multiplied always result in a polynomial of higher degree (unless one of them is a constant polynomial). An example of multiplying polynomials is given below:

Example: Solve (6x−3y)×(2x+5y)

Solution:

⇒ 6x ×(2x+5y)–3y × (2x+5y) ———- Using distributive law of multiplication

⇒ (12x²+30xy) – (6yx+15y²) ———- Using distributive law of multiplication

⇒12x²+30xy–6xy–15y² —————– as xy = yx

Thus, (6x−3y)×(2x+5y)=12x²+24xy−15y²