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Last updated on September 22, 2025

Adding and Subtracting Polynomials

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Algebraic expressions consist of variables, coefficients, and constants combined using basic arithmetic operations. Addition and subtraction are fundamental operations in algebra and follow specific rules for accuracy.

Adding and Subtracting Polynomials for Saudi Students
Professor Greenline from BrightChamps

What are Polynomials?

A polynomial in one variable x can be written in standard form as a0xn + a1xn-1 +... + an. Here, a0, a1,..., an, are real-number coefficients, n is a non-negative whole number, and the powers of x decrease from left to right.

 


For example: 4x3 - 2x2 + 5x + 7 is a polynomial of degree 3 with 4 terms.
 

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What are the Types of Polynomials?

Based on the number of terms, there are 3 types of polynomials. 

 

 

  • Monomial: A polynomial with only one term is called a monomial. Some examples of monomials are: 7x3, -4a2b, 12y

 

  • Binomial: A polynomial with two terms is called a binomial. x2− 9, 3a + 5b, 4m3 + 2 are all examples of binomials.

 

  • Trinomial: A polynomial with three terms is called a trinomial. For example:
    x2 + 3x + 2, a3 − a + 7, 2m2 + 5m + 1

 

Polynomials can have one or more terms and are classified by the number of terms. The degree of a polynomial refers to the highest value of the exponent it has. For instance, in the polynomial 3x2 + 2x - 5, the highest exponent is 2; therefore, it is also the degree of the polynomial. 

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How to Add Polynomials?

There are two rules to be followed when adding polynomials:

 

 

  1. Like terms, which have the same variables and exponents, should always be grouped together. Terms having different variables, exponents, or both are unlike terms.
     
  2. The signs of the terms remain unchanged when adding polynomials.



Let us understand polynomial addition using the following steps to solve an example:

 

Question: Add the polynomials (3x2 - 5x + 2) + (4x2 - 2x + 7)


Solution:

 

Step 1: Arrange the polynomials in standard form
3x2 - 5x + 2 and 4x2 - 2x + 7 are already in standard form.

 


Step 2: Group like terms
(3x2 + 4x2) + (-5x - 2x) + (2 + 7)

 


Step 3: Add the coefficients of like terms
3x2 + 4x2 = 7x2
-5x + (-2x) = -7x
2 + 7 = 9
Answer: 7x2 - 7x + 9
This sum was solved by adding polynomials horizontally.

 

 

We can also do the addition of polynomials vertically. Let us take an example for the same:

 

Question: Add the polynomials (4x2 + 3x + 5) + (2x2 + 6x + 1)
Solution: 


Step 1:Arrange polynomials one below the other and make sure all like terms are aligned together.
                4x²   + 3x   + 5  
              + 2x²   + 6x   + 1  

 


Step 2: Then, calculate the like terms.
To add similar terms, we add the coefficients of the terms and write the variable as is.
4x2 + 2x2. = 6x2
3x + 6x = 9x
5 + 1 = 6
 6x2 + 9x + 6 is the sum of given polynomials.

 

 

How to Subtract Polynomials?

 

The subtraction process of polynomials is similar to the addition process. Addition and subtraction of polynomials can be done two-ways: horizontally and vertically. Two rules to follow when subtracting polynomials are:
 

  1. Like terms must always be grouped.
     
  2. When subtracting, change the signs of all terms in the second polynomial by distributing the minus sign.

     

Let's take an example to understand the steps of polynomial subtraction:

 

Question: Subtract (5x2 + 7x + 2) − (3x2 + 4x - 6)

Solution:
Let’s solve this question using the horizontal method.


Step 1: Arrange polynomials in their standard form (decreasing order of exponents)  and place them next to each other with a subtraction sign between them.
 Since they are already in standard form and placed horizontally,  
(5x2 + 7x + 2) − (3x2 + 4x - 6)
 We can move to the next step.

 

Step 2:Distribute the minus sign to all the terms of the second polynomial
(5x2 + 7x + 2) − (3x2 - 4x + 6)  

 

Step 3: Group like terms,
(5x2 − 3x2) + (7x − 4x) + (2 + 6)

 

Step 4: Calculate
2x2 + 3x + 8
Subtracting (3x2 + 4x - 6) from (5x2 + 7x + 2) gives us the answer 2x2 + 3x + 8.

 

 

Let us solve another example by vertically subtracting the polynomials:

Question: Subtract (6x2 + 5x + 8) − (3x2 + 2x + 4)
Solution:

 

Step 1: Arrange polynomials in standard form. 
The given polynomials are already in their standard form, i.e., written in descending order of exponents.
(6x2 + 5x + 8) − (3x2 + 2x + 4)

 

Step 2: Place polynomials vertically, with like terms aligned one above the other.
   6x²   + 5x   + 8  
- (3x²   + 2x   + 4)

 

Step 3: If any variable terms like x² or x are missing, add a zero coefficient as a placeholder (0x², 0x). Here, we can skip this step since no power terms are missing.

 

Step 4: Change the signs for the second polynomial
 6x²   + 5x   + 8  
-3x2    -2x     - 4

 

Step 5: Calculate
6x2 - 3x2 = 3x2
5x - 2x = 3x
8 - 4 = 4
Therefore, upon subtracting the given terms, we get the answer as: 3x2 + 3x + 4
 

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Real-life Applications of Adding and Subtracting Polynomials

Polynomials are key in solving practical problems across science, engineering, economics, and everyday life. Adding and subtracting polynomials helps predict and model real-world scenarios, including:
 

 

 

  • Measurement calculations in construction and architecture
    Architects use polynomial expressions while calculating the area, perimeter, or volume of irregular shapes in building designs.

     

 

  • Motion and force equations in engineering 
    In kinematics and mechanical systems, position, velocity, and acceleration are modeled using polynomials. Engineers find net forces or combined motions by adding or subtracting these values.

     

 

  • Representing cost, revenue, and profit functions in business 
    Polynomial functions represent cost, revenue, and profit functions over time or production quantity. Businesses use addition or subtraction of these functions for decision-making in regard to sales and restocking.

     

 

  • Describing curves and transformations in animation. 
    Polynomial model curves and transformations for better transitions in computer graphics.

     

 

  • Population modeling for research and mapping
    Polynomial expressions can be used to model population growth for census, pollution levels, or environmental sciences, or resource consumption trends across regions for informed decisions.
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Common Mistakes and How to Avoid Them in Adding and Subtracting Polynomials

Here are some common mistakes that students might make while adding and subtracting polynomials. Let’s see how to avoid them:
 

Mistake 1

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Combining unlike terms
 

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Students may mistakenly combine terms with different variables or exponents. Terms must have the same variables and exponents to be combined. 
For example, adding 3x2 to 5x should result in 3x2 + 5x and not 8x2.

Mistake 2

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Forgetting the distribution of the negative sign

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During subtraction, students often forget to change the signs of the second polynomial. This can be avoided by enclosing the second polynomial in parentheses and distributing the minus sign.
For example, in the subtraction (4x² − 3x + 5) − (2x² + x − 1), distribute the minus sign:
(4x² − 3x + 5) - (2x² + x − 1) = 4x² - 3x + 5 - 2x² - x + 1
Writing it as 4x² - 3x + 5 - 2x² + x - 1 is incorrect.

Mistake 3

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Misaligning terms while using the column method
 

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To avoid misaligning terms, line up the terms by degree and variable before performing operations. For missing terms, add zero as a coefficient to maintain the structure.

Mistake 4

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Ignoring zero coefficients
 

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Students often omit terms with zero coefficients, which may cause confusion during addition or subtraction. It is better to write all terms, including the zero coefficient terms, so there is no confusion or misalignment.

Mistake 5

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Sign errors during simplification
 

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While combining positive and negative coefficients, avoid sign errors by checking each sign carefully.

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Solved Examples of Adding and Subtracting Polynomials

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Problem 1

Add (3x² + 4x + 5) + (2x² − x + 1)

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Okay, lets begin

 5x² + 3x + 6

Explanation

 (3x² + 2x²) + (4x − x) + (5 + 1)

= 5x² + 3x + 6

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Problem 2

Subtract (7x³ + 2x) − (4x³ − 5x)

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Okay, lets begin

3x³ + 7x

Explanation

 7x³ + 2x − 4x³ + 5x = (7x³ − 4x³) + (2x + 5x) = 3x³ + 7x

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Problem 3

Add given polynomials using vertical addition (4x² + 6x + 3) + (x³ + 2x + 5)

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Okay, lets begin

x³ + 4x² + 8x + 8

Explanation

 4x² + 6x + 3
                     x³ + 2x + 5
                   —-------------------
                   x³ + 4x² + 8x + 8

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Problem 4

Subtract the polynomials (5x²y − 3xy² + 7) − (2x²y + 4xy² − 2)

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Okay, lets begin

3x²y - 7xy² + 9

Explanation

Step 1: Distribute negative sign, 5x²y − 3xy² + 7 − 2x²y − 4xy² + 2

 

Step 2: Group like terms: (5x²y − 2x²y) + (−3xy² − 4xy²) + (7 + 2)

 

Step 3: Simplify: 3x²y - 7xy² + 9

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Problem 5

Add the polynomials, (−2x³ + x² − 4x + 6) + (x³ − 5x² + 3x − 1)

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Okay, lets begin

-x³ - 4x²  - x + 5

Explanation

Step 1: Group the like terms
(−2x³ + x³) + (x²  − 5x² ) + (−4x + 3x) + (6 − 1)

 

Step 2: Add the coefficients of each group: 
-2x³ + x³ = -x³
x² - 5x² = -4x² 
-4x + 3x = -x
6 - 1=5.

Hence, the answer is −x³ − 4x² − x + 5

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FAQs on Adding and Subtracting Polynomials

1.How do we determine the degree of polynomials?

The degree of a polynomial is the highest exponent of any term. For example, 4x3 + 3x2 + 9x + 6, the term 4x3  has the highest power, which is 3. So, 3 is the degree of this polynomial.
 

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2.What are like terms?

Terms having the same variables and exponents are known as like terms.
 

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3.What are the methods used for addition and subtraction of polynomials?

Addition and subtraction of polynomials can be carried out using two methods: vertical method and horizontal method.
 

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4.Can terms having different exponents be combined?

No, only like terms can be combined.
 

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5.What are the types of polynomials based on degree?

Name of polynomial

Degree

Constant 

0                   Example: 7, -2

Linear

1                   Example: 2x + 5, -x

Quadratic

2                   Example: x2

Cubic

3                   Example: x3 - 2x2 + x

Quartic

4                   Example: 6x4 

Quintic

5                   Example: 3x5 + 2x2 

Higher Degree

6 or more      Example: x6 + x3 - 1

 

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6.How does learning Algebra help students in Saudi Arabia make better decisions in daily life?

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

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7.How can cultural or local activities in Saudi Arabia support learning Algebra topics such as Adding and Subtracting Polynomials?

Traditional games, sports, or market activities popular in Saudi Arabia can be used to demonstrate Algebra concepts like Adding and Subtracting Polynomials, linking learning with familiar experiences.

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8.How do technology and digital tools in Saudi Arabia support learning Algebra and Adding and Subtracting Polynomials?

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

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9.Does learning Algebra support future career opportunities for students in Saudi Arabia?

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

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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.

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Fun Fact

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

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