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 be placed 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:
 

Example: 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 \(6 + 3x^2 - 5x \) from \(-2x + 4 - x^2 \)
 

Step 1: Arrange polynomials in standard form
 

First polynomial : \(-2x + 4 - x^2 \) → \(-x^2 - 2x + 4 \)

Second polynomial: \(6 + 3x^2 - 5x \) → \(3x^2 - 5x + 6 \)

Step 2: Arrange like terms vertically
 

\(-x^2 - 2x + 4 \)

\(-(3x^2 - 5x + 6) \)
 

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
 

\(-x^2 - 2x + 4 \)

\(-3x^2 + 5x - 6 \)

Step 5: Calculate
 

\((-x^2 - 3x^2) = -4x^2 \)

\((-x^2 - 3x^2) = -4x^2 \)

\((4 - 6) = -2 \)
 

Therefore, subtracting the given polynomials vertically, we get the answer: \(-4x^2 + 3x - 2 \).
 

Subtracting polynomials is an important skill in algebra. It helps students make expressions simpler and solve equations more easily. It may seem tricky at first, but learning the right steps and avoiding common mistakes makes it much easier. Being good at this also helps with more advanced math later.

• Use parentheses when subtracting to keep track of which polynomial you are removing.
   
• Distribute the negative sign to every term in the polynomial being subtracted to avoid mistakes.
   
• Combine like terms carefully, making sure to only combine terms with the same variable and exponent.
   
• Reorder terms if necessary to make it easier to see and subtract like terms.
   
• Pay close attention to signs, ensuring positives and negatives are handled correctly during subtraction.

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: Polynomials can represent dimensions; subtraction helps find usable areas after accounting for structures like columns or staircases.
   
• 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.