Adding and Subtracting Polynomials

A polynomial in one variable x can be written in standard form as \(a_0x^n + a_1x^{n-1} +... + a_n\). Here, a₀, a₁,..., a_n, are real-number coefficients, n is a non-negative whole number, and the powers of x decrease from left to right.

For example: \(4x^3 - 2x^2 + 5x + 7\) is a polynomial of degree 3 with 4 terms.
 

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: \(7x^3, -4a^2b, 12y\).

• 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. 

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 \((3x^2 - 5x + 2) + (4x^2 - 2x + 7)\)

Solution:

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

Step 2: Group like terms

\((3x^2 + 4x^2) + (-5x - 2x) + (2 + 7)\)

Step 3: Add the coefficients of like terms

\(3x^2 + 4x^2 = 7x^2\)

\(-5x + (-2x) = -7x \)

\(2 + 7 = 9\)

Answer: \(7x^2 - 7x + 9\)

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 \((4x^2 + 3x + 5) + (2x^2 + 6x + 1)\)

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.

\(4x^2 + 2x^2. = 6x^2\)

\(3x + 6x = 9x\)

\(5 + 1 = 6\)

 \(6x^2 + 9x + 6\) 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 \((5x^2 + 7x + 2) − (3x^2 + 4x - 6)\)

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,  

\((5x^2 + 7x + 2) − (3x^2 + 4x - 6)\)

 We can move to the next step.

Step 2:Distribute the minus sign to all the terms of the second polynomial.

\((5x^2 + 7x + 2) − (3x^2 - 4x + 6)\)  

Step 3: Group like terms,

\((5x^2 − 3x^2) + (7x − 4x) + (2 + 6)\)

Step 4: Calculate:

\(2x^2 + 3x + 8\)

Subtracting \((3x^2 + 4x - 6)\) from \((5x^2 + 7x + 2)\) gives us the answer \(2x^2 + 3x + 8\).

Let us solve another example by vertically subtracting the polynomials:

Question: Subtract \((6x^2 + 5x + 8) − (3x^2 + 2x + 4)\)

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.

\((6x^2 + 5x + 8) − (3x^2 + 2x + 4)\)

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.

  \(6x²   + 5x   + 8 \) 
\(-3x^2    -2x     - 4\)

Step 5: Calculate:

\(6x^2 - 3x^2 = 3x^2\)

\(5x - 2x = 3x\)

\(8 - 4 = 4\)

Therefore, upon subtracting the given terms, we get the answer as: \(3x^2 + 3x + 4\)
 

Adding and subtracting polynomials are the basics in algebra, and students can follow these tips and tricks to master this concept easily: 

• Line up like terms first: Always rewrite your polynomials so that terms with the same variable and exponent are aligned. This helps you easily see which terms you can combine.
  
   
• Pay close attention to the signs when subtracting: For subtraction, make sure you distribute the minus sign across each term of the second polynomial before you combine like terms.
  
   
• Choose the method that works for you (horizontal vs vertical): Horizontal method: Write the two polynomials in one line, for example, (3x2−5x+2)+(4x2−2x+7) and then combine like terms.
  
  Vertical method: Write one polynomial above the other, align like terms in columns, then add/subtract. Especially helpful for longer polynomials.
  
   
• Use a placeholder (0 ⋅ …) for missing terms: If one polynomial lacks a term that the other has For example, one has a \(𝑥^2\) term, the other doesn’t, it’s helpful to write \(0𝑥^2\) or 0𝑥 so you can align everything cleanly.
  
   
• Simplify step-by-step and double-check coefficients: After aligning and distributing signs, carefully add/subtract the coefficients of like terms. Then check that the variables and exponents are correct in your final expression. Rushing often leads to sign errors or combining wrong terms.

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 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.