12.6: Subtracting Polynomials
Have you ever thought about the area of a pool? Take a look at this dilemma.
A concrete walkway surrounds a rectangular swimming pool. In order to know how much weather treatment to buy, the owner must know how many square units of concrete he has. The walkway is 5 feet wide on all sides. The swimming pool has a length of
In order to find the area of the concrete, we must find the area of the large rectangle and then subtract the area of the swimming pool.
Do you know how to do this? To accomplish this task, you will need to understand how to subtract polynomials. You will learn how to do this in this Concept.
Guidance
A polynomial is an algebraic expression that shows the sum of monomials.
Since the prefix mono means one, a monomial is a single piece or term. The prefix poly means many. So the word polynomial refers to one or more than one term in an expression. The relationship between these terms may be sums or difference.
Polynomials:
We call an expression with a single term a monomial, an expression with two terms is a binomial, and an expression with three terms is a trinomial. An expression with more than three terms is named simply by its number of terms—“fiveterm polynomial.”
Just like we could add polynomials, we can subtract them too. We can perform this operation both vertically and horizontally. Let’s start with vertically.
When we subtract polynomials, we can use a similar procedure as with addition—we can subtract vertically. However, remember that subtraction is the same as “adding the opposite.” In other words,
Remember that subtracting is the same as adding the opposite. Write this down in your notebook.
Take a look at this one.
Set up the problem vertically by aligning the like terms.
When you add the opposite, the sign changes on each of the terms in the subtracted polynomial. Inside the parentheses, the coefficient of
Now we can look at subtracting polynomials horizontally.
When we added polynomials, we used two methods—adding vertically and adding horizontally. You just learned to subtract polynomials vertically. As you have guessed, we can also subtract polynomials horizontally. First we will review the distributive property.
The distributive property: For all real numbers
Remember to be careful with negative signs when using distributive property.
Now, let us remember that coefficients are the numerical factors of variables. The coefficient of
Take a look at this one.
Then take a look at this one.
Here you can insert the 1 and then multiply. As with adding the opposite, the sign changes on each of the terms in the polynomial.
We can now use this method to subtract polynomials horizontally. First we’ll distribute the negative sign to each of the terms in the subtracted polynomial and then we will combine like terms just as we did when we added polynomials.
Here is another one.
It may seem that way, but if you go step by step and remember that subtracting is adding the opposite, then you will be able to subtract polynomials vertically and horizontally.
Subtract the following polynomials.
Example A
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
The length of the large rectangle measures
So its area will be
The area of the swimming pool will be its length times its width or
Area of the swimming pool is
To find the area of the concrete, subtract the area of the swimming pool from the total area:
This is the answer.
Vocabulary
 Polynomial
 one or more terms in an expression, often referred to specifically in situations where there are more than three terms.
 Like Terms
 terms that have the same variable and power.
 Area
 the space inside an object or area. It is measured in square units.
Guided Practice
Here is one for you to try on your own.
Solution
This is the answer.
Video Review
Practice
Subtract the following polynomials vertically.

(6x2+5x)−(3x2−14x+2) 
(3x2+5x+3)−(2x2−x+4) 
(5xy+5x+3)−(12xy−4x−8) 
(5y2+5y−2)−(3y2−6y+5) 
(8x+5y+1)−(9x+2y+5) 
(7x2+x−3)−(3x2+3x+4) 
(8x+5y+4)−(3x−9y−5) 
(18x3+2x2+8x+2)−(3x2−4x−9) 
(8x+9y−20)−(3x−14) 
(16x2+5x−3y+7)−(3x−14y+10) 
(18x2+5xy−6x+21)−(3x2−14xy−9x+1) 
(7y3+4y2−3y−1)−(y3+6y2−4)
Subtract the following polynomials horizontally.
 \begin{align*}(m^2+17m11)(3m^2+8m+12)\end{align*}
 \begin{align*}(z^2+3z)(3z^2+7z+16)(4z13)\end{align*}
 \begin{align*}(5x^2+3xy)(3x^2+7xy+6)(4xy13)\end{align*}
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Term  Definition 

Area  Area is the space within the perimeter of a twodimensional figure. 
like terms  Terms are considered like terms if they are composed of the same variables with the same exponents on each variable. 
Polynomial  A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents. 
Image Attributions
Here you'll subtract polynomials vertically and horizontally.