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.

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 + 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.