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.

Examples of polynomials: (x + 2x) + (x - 2), (2x) - (x + 1), and (4x³ - 5x) + (3x + 6).

Standard Form of a Polynomial

a₀xⁿ+a₁x^n-1+a₂x^n-2+……..+a_nx⁰

• a₀, a₁, a₂, a₃,…. a_n are constants.
• x is a variable
• n is any whole number

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. 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. 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 ones last.
• Step 2: Group the like terms, i.e., variables having the same power/exponent.
• Step 3: Perform calculations.

Example: Perform addition between polynomials 3x + 2x + 1 and 4x + 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 + 7x³ - 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 ones last.
• Step 2: Group the like terms, i.e., variables having the same power/exponent.
• Step 3: Signs of subtracting polynomials get changed, 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 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 polynomials get changed, 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

Also Check

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