BrightChamps Logo
Login

Summarize this article:

Live Math Learners Count Icon181 Learners

Last updated on 19 September 2025

Subtracting Polynomials

Professor Greenline Explaining Math Concepts

Subtracting polynomials involves changing the signs of terms in one polynomial before combining like terms. Depending on the expressions, it is similar to the addition of polynomials. The positive signs must be changed to negatives and vice versa.

Subtracting Polynomials for UAE Students
Professor Greenline from BrightChamps

What is Subtracting Polynomials?

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:

 

  1. Like terms should always be together.
     
  2. The signs of all terms being subtracted must change, i.e., all terms having negative signs changes to positive and vice versa.
Professor Greenline from BrightChamps

What are the Steps to Subtract Polynomials?

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
 

  1. Ensure the polynomials are in their standard form.
     
  2. Place the polynomials next to each other.
     
  3. Change the signs for all terms in the parentheses. Do this for the second polynomial.
     
  4. Separate and arrange like terms together.
     
  5. Calculate the terms to determine the result.
     

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
 

  1. Arrange polynomials in their standard form.
     
  2. Place them vertically. The like terms are placed one above the other.
     
  3. In the case of missing terms, use 0 as a coefficient to maintain alignment in the equation. 
     
  4. Change signs of terms in the second polynomial.
     
  5. Calculate
     

For example: Subtract 6 + 3x–  5x from –2x + 4 – x2
 

Step 1: Arrange polynomials in standard form
 

First polynomial : –2x + 4 – x² → –x² – 2x + 4

Second polynomial: 6 + 3x– 5x → 3x² – 5x + 6


Step 2: Arrange like terms vertically
 

           –x2 – 2x + 4

         –(3x– 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
 

Professor Greenline from BrightChamps

Real-life Applications of Subtracting Polynomials

We subtract polynomials to simplify and solve expressions. Subtracting polynomials can also be used to solve real-life situations like:

 

  1. Calculating usable floor area in architecture

    Polynomials can represent dimensions; subtraction helps find usable areas after accounting for structures like columns or staircases.
     
  2. 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.
     
  3.  Relative velocity in physics

    In motion-related problems, relative velocity is found using polynomial subtraction of velocities of two objects.
     
  4. 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.
     
  5. 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.
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Subtracting Polynomials

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.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not changing signs of the second polynomial

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Do not forget that the sign of each term of the second polynomial must be changed.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Combining unlike terms

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Combine like terms, as unlike terms (x2 and x) cannot be combined.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Errors in rearranging

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Pay close attention while writing polynomials in their standard forms and while changing the signs.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Dropping negative signs

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Make sure you don't miss any negative signs while performing calculations.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Miscalculating arithmetic with coefficients

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Do not rush with calculations, check for all signs and coefficients.

arrow-right
Max from BrightChamps Saying "Hey"
Hey!

Solved Examples of Subtracting Polynomials

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Subtract (7x + 4) - (3x - 2)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

4x + 6

Explanation

First, distribute the negative sign from the second polynomial 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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

Subtract (6 + 3x² − 5x) from (−2x + 4 − x²)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

−4x2 + 3x − 2

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

Subtract (5x³ + 2x²− 4x + 6) − (3x − x² + x−1)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

2x3 + 3x2 − 5x + 7

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Subtract (8x³ + 2 − x) − (5x³ + 4x)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

3x3 − 5x + 2

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 5

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)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

P(x) = x2 + 5x + 40

Explanation

P(x) = (4x2 + 10x + 100) − (3x2 + 5x + 60)

Change signs: 4x2 + 10x + 100 − 3x2 − 5x − 60

Combine like terms: x2 + 5x + 40

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

FAQs on Subtracting Polynomials

1.What property is used when subtracting polynomials?

While subtracting polynomials, the distributive property is used across the terms of the second polynomial.

Math FAQ Answers Dropdown Arrow

2.How do you simplify the process of subtracting polynomials?

Subtraction of polynomials is a laborious process. However, it can be simplified by using the below-mentioned steps:

 

  • Rearrange the polynomials to get the standard form.
     
  • Use the distributive property and assign a negative sign to all terms in parentheses of the second polynomial.
     
  • Combine like terms
     
  • Simplify the result by adding/subtracting coefficients.

Math FAQ Answers Dropdown Arrow

3.Can we subtract polynomials with different degrees?

Yes, one can do so by aligning like terms and subtracting where possible.

Math FAQ Answers Dropdown Arrow

4.What happens when a variable or exponent is missing?

For a missing term, use zero as the coefficient. This helps in maintaining alignment during subtraction.

Math FAQ Answers Dropdown Arrow

5.What is the difference between the horizontal and vertical method?

Horizontal method Vertical method
The polynomials are written in the same line, right next to each other. Polynomials are written one above another.
Steps include: Distributing the negative sign, then combining like terms. Steps involved: Align like terms in columns, change signs of the second row, then subtract
Preferred for short and simple expressions. Used in longer and more complex polynomial subtractions having multiple terms.
It is visually less clear and has a risk of misalignment. It provides visually clear alignments of terms, making subtraction easier.
The chances of making sign mistakes are comparatively higher. The chances of sign errors are low due to clarity in alignment.

 

The horizontal method can be used for quick solutions in simple polynomial subtraction. Subtracting polynomials vertically is better when used for complex polynomials, as it ensures greater accuracy.

Math FAQ Answers Dropdown Arrow

6.How does learning Algebra help students in United Arab Emirates make better decisions in daily life?

Algebra teaches kids in United Arab Emirates to analyze information and predict outcomes, helping them in decisions like saving money, planning schedules, or solving problems.

Math FAQ Answers Dropdown Arrow

7.How can cultural or local activities in United Arab Emirates support learning Algebra topics such as Subtracting Polynomials?

Traditional games, sports, or market activities popular in United Arab Emirates can be used to demonstrate Algebra concepts like Subtracting Polynomials, linking learning with familiar experiences.

Math FAQ Answers Dropdown Arrow

8.How do technology and digital tools in United Arab Emirates support learning Algebra and Subtracting Polynomials?

At BrightChamps in United Arab Emirates, we encourage students to use apps and interactive software to demonstrate Algebra’s Subtracting Polynomials, allowing students to experiment with problems and see instant feedback for better understanding.

Math FAQ Answers Dropdown Arrow

9.Does learning Algebra support future career opportunities for students in United Arab Emirates?

Yes, understanding Algebra helps students in United Arab Emirates develop critical thinking and problem-solving skills, which are essential in careers like engineering, finance, data science, and more.

Math FAQ Answers Dropdown Arrow
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

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.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom