Last updated on July 5th, 2025
A method where the polynomials are subtracted by converting signs to opposites is known as subtracting polynomials. Depending on the expressions, it is similar to addition of polynomials. The positive signs must be changed to negatives and vice versa.
Subtraction of polynomials is done using two methods, vertical and horizontal. For simplification, like terms in a polynomial are separated and aligned together. Columns help in matching correct terms during complicated subtractions and this is especially done in the vertical method.
Two rules while subtracting polynomials are:
Polynomials are subtracted using either the vertical method or the horizontal method. In the horizontal method of polynomial subtraction, the signs of terms that are in parentheses in the second expression change. This simplifies the subtraction allowing it to be solved as an addition. In the vertical method, the polynomials are arranged in columns one above another based on like terms. The signs are then changed accordingly, and subtraction is carried forward.
Let's understand the steps used in both methods:
Method 1: Horizontal method
Let's apply these steps to an example:
Question: Subtract 2a + 7b -3c from 6a - 4b + 5c
As the polynomials are already in their standard form, let’s start with step 2, which is to place them horizontally.
(6a - 4b + 5c ) - ( 2a + 7b - 3c )
Step 3: Change signs
6a - 4b + 5c - 2a - 7b + 3c
Step 4: Arrange like terms together
6a - 2a - 4b - 7b + 5c + 3c
Step 5: Solve the expression
4a - 11b + 8c
Therefore, upon subtraction 2a + 7b -3c from 6a - 4b + 5c, we get 4a - 11b + 8c.
Method 2: Vertical method
For example: Subtract 6 + 3x2 - 5x from -2x + 4 - x2
Step 1: Arrange polynomials in standard form
First polynomial : -2x + 4 - x² → -x² - 2x + 4
Second polynomial: 6 + 3x2 - 5x → 3x² - 5x + 6
Step 2: Arrange like terms vertically
-x2 - 2x + 4
-(3x2 - 5x + 6)
Step 3: Since both polynomials have terms for x2, we don't need to use 0 as a coefficient in this case.
Step 4: Change signs
-x2 - 2x + 4
-3x2 + 5x - 6
Step 5: Calculate
(-x² - 3x²) = -4x²
(-2x + 5x) = 3x
(4 - 6) = -2
Therefore, subtracting the given polynomials vertically, we get the answer: -4x2 + 3x - 2
We subtract polynomials to simplify and solve expressions. Subtracting polynomials can also be used to solve real-life situations like:
Calculating usable floor area in architecture
Polynomial subtraction is used to calculate areas. Architects find the usable floor area of staircases, columns, etc., to plan a building's layout accordingly.
Calculating net profit in a business
Business models like revenue and cost have changing values that are modelled as polynomials. The cost model can be subtracted from revenue to find net profit. This process requires polynomial subtraction.
Relative velocity in physics
In motion-related problems, relative velocity is found using polynomial subtraction of velocities of two objects.
Calculating drug concentration in a patient’s body
Pharmacologists use polynomial subtraction to find portions of drugs that have been metabolized in a patient's body over time. This is required for safe drug dosages at regular intervals.
Stock tracking in inventory management
The number of items in stock and sold is expressed as polynomials. Retailers use polynomial subtraction to determine stock levels and make restocking decisions.
Performing algebraic operations with polynomials can be a little confusing in the beginning. Here are a few common errors related to subtraction of polynomials and how they can be avoided.
Subtract (7x + 4) - (3x - 2)
4x + 6
First, we distribute the negative sign and remove the brackets.
7x + 4 − 3x + 2
Then, we combine the like terms
= (7x − 3x) + (4 + 2)
So, (7x + 4) - (3x - 2) = 4x + 6
Subtract (6 + 3x² − 5x) from (−2x + 4 − x²)
−4x2 + 3x − 2
Step 1: Arrange the polynomials in their standard form
−x2 − 2x + 4
3x2 − 5x + 6
Step 2: subtract
(−x2 − 2x + 4) − (3x2 − 5x + 6)
Change signs: −x2 −2x + 4 − 3x2 + 5x − 6
Combine like terms: (−x2 − 3x2) + (−2x + 5x) + (4 − 6) = −4x2 + 3x − 2
Subtract (5x^3 + 2x²− 4x + 6) − (3x^3 − x² + x−1)
2x3 + 3x2 − 5x + 7
Distribute the minus sign and remove brackets:
(5x3 + 2x2 − 4x + 6) − (3x3 + x2 − x + 1)
Group like terms:
(5x3 -3x3) + (2x2 + x2) + (-4 - x) + (6 + 1)
Simplify all terms:
2x3 + 3x2 - 5x + 7
Subtract (8x^3 + 2 − x) − (5x^3 + 4x)
3x3 − 5x + 2
Since neither polynomial has an x2 term, use 0 as the coefficient for the missing term i.e., x2:
(8x3 − x + 0x2 + 2) − (5x3 + 4x + 0x2 + 0)
Distribute minus: 8x3 − x + 2 − 5x3 − 4x
Combine the terms: (8x3 − 5x3) + (−x − 4x) + 2 = 3x3 − 5x + 2
A company's revenue and cost polynomials are: Revenue: R(x) = 4x² + 10x + 100 Cost: C(x) = 3x² + 5x + 60 Find the profit polynomial P(x) = R(x) - C(x)
P(x) = x2 + 5x + 40
P(x)=(4x2 + 10x + 100) − (3x2 + 5x + 60)
Change signs: 4x2 + 10x + 100 − 3x2 − 5x − 60
Combine like terms: x2 + 5x + 40
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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