Adding and Subtracting Polynomials

Polynomials are algebraic expressions composed of variables and coefficients, combined through addition, subtraction, and multiplication. Among the fundamental operations in polynomials, addition and subtraction require specific rules to ensure accuracy and consistency.

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Standard Form of a Polynomial

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 of polynomials-
• x³ + 2x² + x - 2,
• 2x² - x + 1,
• 4x³ - 5x² + 3x + 6.

Read More: Polynomials

Key Points to Remember

1. Like Terms: Always group and operate on terms that have the same variable and exponent.

• Example: 2x² and 5x² can be combined, but x² and x cannot.

2. Standard Form: Arrange polynomials in descending order of exponents.

3. Addition Rules:

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 remain the same.

4. Subtraction Rules:

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 +.

Methods of Addition and Subtraction

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.

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

Solution:

Group like terms:
3x² + 4x² + 2x + x + 1 + 9

Compute:
(3 + 4)x² + (2 + 1)x + 10
7x²+3x+10

Adding Polynomials Vertically

Steps to Add

• Step 1: Write polynomials in standard form, and align like terms in columns.
• Step 2: Perform calculations.

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

Solution:

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

3x³ - 2x² + 0x + 1
4x³ + 7x² - 1x + 9

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(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.

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

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.

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

-    -      -      +    -

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(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

-    -     +

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(4 - 2)a + (-4-3)b + (1 + 1)c
2a - 7b + 2c

Related Reads:

• Multiplying Polynomials
• Dividing Polynomials – Long Division
• Dividing Polynomials - Synthetic Division