Addition and Subtraction of Polynomials

Different operations cab be performed on the polynomials like addition, subtraction, multiplication, and division. A polynomial is an expression within which a finite number of constants and variables are combined using addition, subtraction, multiplication, and exponents. Adding and subtracting polynomials is just adding and subtracting their like terms. The sum of two monomials is called a binomial and the sum of three monomials is called a trinomial. The sum of a finite number of monomials in x is called a polynomial in x. The coefficients of the monomials in a polynomial are called the coefficients of the polynomial. If all the coefficients of a polynomial are zero, then the polynomial is called the zero polynomial. Several operations can be done with polynomials.

Like Terms

These are the terms whose variables as well as the exponent are the same.

Example: x -2x 7x are all the like terms because the variables are all x.

Example: x -2y 5xy aren’t like terms because the variables are different.

Addition of polynomials

Two polynomials can be added by using arithmetic operators plus (+) or minus(-). Adding polynomials is simply “combining like terms” and then add the like terms.

Steps for Addition:

Arrange the Polynomial in standard form that is the term with the highest degree is first.
Arrange the like terms together.
Add the like terms.

Example 1: Add the expression 5x+2x

Solution: Both 5x and 2x involve a single variable x, each having the same exponent (=1). Hence the two terms can be simply added as indicated by the plus (+) sign. So, 5x + 2x = (5 + 2)x = 7x.

Example 2: Add the expression $4x^{2}+6x+7$ and $2x^{2}-2x-1$

Solution: Step 1: Start with $4x^{2}+6x+7 + 2x^{2}-2x-1$

Step 2: Write like terms together $4x^{2}+2x^{2}+6x-2x+7-1$

Step 3 : Add the like terms $(4+2)x^{2}+(6-2)x+(7-1)$ = $6x^{2}+4x+6$ .

Example 3: Add the expression $10x^{2}+8x-6$ and $4x^{2}+3x$ .

Solution: Step 1: Start with $10x^{2}+8x-6 + 4x^{2}+3x$ There is no like term for -6 in another polynomial, so we don’t have to add anything to it.

Step 2: Write like terms together $10x^{2}+4x^{2}+8x+3x+(-6)$

Step 3: Add the like terms $(10+4)x^{2}+(8+3)x+(-6)$ = $14x^{2}+11x-6$

Subtraction of polynomials

Two polynomials can be subtracted by using arithmetic operators plus (+) or minus(-). The operations of subtraction with polynomials is the same as the operations of addition with polynomials. Subtraction polynomials are simply “adding the opposite”.

Steps for Subtraction:

Enclose the part of the expression to be deducted in parentheses with a negative (-) sign prefixed.
Remove the parentheses by changing the sign of each term of the polynomial expression.
Arrange the like terms.
Add the like terms to find the required subtraction.

Example 1: Subtract: 5x – 7y + 3z from 6x + 8y – 5z.

Solution: Step 1: We need to enclose the first part which is to be subtracted in parentheses with a negative (-) sign prefixed 6x+8y-5z – (5x-7y+3z)

Step 2: Remove the parentheses and change the sign of each term 6x+8y-5z – 5x+7y-3z

Step 3: Arrange the like terms 6x-5x+8y+7y-5z-3z

Step 4: Add the like terms x+15y-8z

Example 2: Solve (15xy + 2zx) – (10xy + 4zx)

Solution: Step 1: 15xy+2zx – (10xy +4zx)

Step 2: 15xy+2zx-10xy-4zx

Step 3: 15xy-10xy+2zx-4zx

Step 4: 5xy-2zx