Adding and Subtracting Polynomials (original) (raw)
Last Updated : 27 Apr, 2026
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
**a 0 x n +a 1 x n-1 +a 2 x n-2 +……..+a n x 0
- a0, a1, a2, a3,…. an 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:
- Always take the like terms together while performing addition/subtraction. Example: 2x & 5x, 3x2 & 7x2.
- Signs of all terms in polynomials remain the same.
**4. Subtraction Rules:
- Always take the like terms together while performing addition/subtraction. Example: 4x2 & 10x2, 6x3 & 7x3.
- 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 + 9Compute:
(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+7x 2 +1 and 2x 2 -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