Adding and Subtracting Polynomials

While performing addition or subtraction between polynomials, we need to remember that we should add or subtract the terms having the same power. The power of a variable should be whole numbers but not negative and irrational numbers. Let’s look at the standard form of polynomial constants and variables-

a₀xⁿ+a₁x^n-1+a₂x^n-2+……..+a_nx⁰

Where

a₀, a₁, a₂, a₃,…. a_n are constants.

x is a variable

n is any whole number

Examples for polynomials– x³+2x²+x-2, 2x²-x+1, and 4x³-5x²+3x+6.

Rules for Addition

1. Always take the like terms together while performing addition/subtraction. Like terms are the constants with variables having the same power/exponent. Example: 2x & 5x, 3x² & 7x².
2. Signs of all terms in polynomials remains the same.

Rules for Subtraction

1. Always take the like terms together while performing addition/subtraction. Like terms are the constants with variables having the same power/exponent. Example: 4x² & 10x², 6x³ & 7x³.
2. Signs of all terms of a subtracting polynomial will get changed i.e., + changes to – and – changes to +.

We can perform addition/subtraction between polynomials in two ways. Either horizontally or vertically.

Adding Polynomials Horizontally

Before moving toward steps to perform addition, we need to remember the above-specified rules first.

Steps to Add

Step 1: Arrange the polynomial in standard form i.e., arrange the polynomial in such a way that terms with variables having higher exponents are arranged first and lower at last.

Step 2: Group the like terms i.e., variable having the same power/exponent.

Step 3: Perform calculations.

Let’s look into an example 

Example: Perform addition between polynomials 3x²+2x+1 and 4x²+x+9.

Solution:

Step 1: Arrange polynomial in standard form.

The given two polynomials are already in standard forms.

Step 2 and 3: Group like terms and perform calculation.

(3x²+2x+1)+(4x²+x+9)= 3x²+4x²+2x+x+1+9

                                    = (3+4)x²+(2+1)x+10

                                    = 7x²+3x+10

Adding Polynomials Vertically

Rules need to remember first.

Steps to Add

Step 1: Arrange both the polynomials one above the other with like terms placed one above the other in standard form. And if any of the polynomials didn’t have the variable with the same exponent as the above polynomial use 0 as a coefficient to avoid confusion.

Step 2: Perform calculations.

Let’s look into an example

Example: Perform addition between polynomials 3x³-2x²+1 and 4x³+7x²-x+9.

Solution:

Step 1&2: Arrange polynomial one above the other in standard form and perform calculations.

3x³-2x²+0x+1

4x³+7x²-1x+9

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(3+4)x³+(-2+7)x²+(0-1)x+1+9

7x³+5x²-x+10

Subtracting Polynomials Horizontally

Before moving toward steps to perform subtraction, we need to remember the above-specified rules first.

Steps to Subtract

Step 1: Arrange the polynomial in standard form i.e., arrange the polynomial in such a way that terms with variables having higher exponents are arranged first and lower at last.

Step 2: Group the like terms i.e., variable having the same power/exponent.

Step 3: Signs of subtracting polynomial get’s changes i.e., from + to – and – to +.

Step 4: Perform calculations.

Let’s look into an example

Example: Perform subtraction between polynomials x+7x²+1 and 2x²-7.

Solution:

Step 1: Arrange polynomial in standard form.

7x²+x+1 and 2x²+0x-7

Step 2: Group like terms and Signs of subtracting polynomial get’s changed and calculate the result

(7x²+x+1)-(2x²+0x-7)= (7-2)x²+(1-0)x+(1+7)

                                  = 5x²+x+8

Subtracting Polynomials Vertically

Rules need to remember first.

Steps to Subtract

Step 1: Arrange both the polynomials one above the other with like terms placed one above the other in standard form. And if any of the polynomials didn’t have the variable with the same exponent as the above polynomial use 0 as a coefficient to avoid confusion.

Step 2: Signs of subtracting polynomial get’s changes i.e., from + to – and – to +.

Step 3: Perform calculations.

Let’s look into an example

Example 1: Perform Subtraction between polynomials 5x³+5y²-2z²+1 and 4x³+y²-x+2.

Solution:

Arrange polynomial one above the other in standard form and perform subtraction.

5x³+5y²-2z²+0x+1

4x³+1y²+0z²-1x+2

                                                                             –    –      –      +    –

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(5-4)x³+(5-1)y²+(-2-0)z²+(0+1)x+(1-2)

x³+4y²-2z²+x-1

Example 2: What is the resultant polynomial if we perform a subtraction between two polynomials 4a-4b+c and 2a+3b-c

Solution:

Arrange polynomials one above the other in standard form and perform subtraction

4a-4b+1c

2a+3b-1c

                                                                                   –    –     +

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(4-2)a+(-4-3)b+(1+1)c

2a-7b+2c

These are the ways to perform addition and subtraction between polynomials.