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Subtracting Polynomials

Introduction

Before starting this lesson, it is highly recommended that you do the lesson on adding polynomials.  These two lessons are often presented as a single lesson in a mathematics course and are meant to be done together in order.

In order to be successful at this lesson, you must have a good understanding of the rules of subtraction as well as how to add polynomials.  Remember that subtraction can also be thought of as “adding the opposite.”

Subtracting_Polynomials_vis_1

 

Furthermore, subtracting a negative is the same as adding a positive.

Subtracting_Polynomials_vis_2

The same rules apply when adding and subtracting polynomials.  Remember to “add the opposite” when subtracting each term in a polynomial.

 

Lesson

An expression containing algebraic terms is called a polynomial.  In the lesson on adding polynomials, you learned that adding polynomials simply involves adding individual terms of the polynomials together.  A problem is given below.


Subtracting_Polynomials_vis_3The work for this problem has been done using vertical addition.  As you become more comfortable adding (and subtracting) polynomials, you will be able to add or subtract like terms by categorizing them and doing the problem horizontally.  The rest of the problems in this lesson done by showing the work horizontally.

 

 

When subtracting polynomials, the minus sign must be distributed to each term of the polynomial that follows it.


Subtracting_Polynomials_vis_4

 

 

In the problem above the x terms are underlined and the constant terms are double underlined.  This makes it easier to find the terms that can be combined.  When dealing with larger polynomials, you may need to come up with several different ways to identify each term.  Many students have success using colored pencils to differentiate between terms.

Consider the original (addition) problem at the top of this page.  Notice what happens in a similar problem where you subtract the second polynomial from the first.

Subtracting_Polynomials_vis_5

 

The red rainbows above represent the fact that the subtraction sign is distributed to each of the terms in the second polynomial.  This strategy should be used for every problem in which a polynomial is being subtracted.  Showing your work is critical here… doing 2 or more operations in your head should generally be avoided because the chance of making a mistake grows as the polynomials become larger.  The #1 cause of mistakes is doing too much mental math instead of writing down all your work.  Make sure you write down all your steps and give yourself the best chance to get it right.  An example of good work on a more difficult problem is shown below.


Subtracting_Polynomials_vis_6

 

Now it’s your turn.  Try the following problem.  When you are done, check your answer (and especially your work).

Problem:

(5x4 – 8x3 – 15x2 + 10x – 15) – (4x4 – 5x3 + 4x2 – 13x + 11)

Solution:

 Subtracting_Polynomials_vis_7a

 

The following problem contains both addition and subtraction.  Take your time and show all your work and you should be able to do it correctly.  Again, check your answer and work when finished.

Problem: (2x2 – 4x + 11) – (8x2 – 3x + 14) + (3x2 + 6x + 5)

Solution:  

Subtracting_Polynomials_vis_8a

 

 

Try It

1)  (5x2 – 3x + 22) + (6x2 + 14x + 21)

2)  (4x3 – 3x2 + 14x + 18) – (-4x3 – 6x2 + 9x – 8)

3)  (8x5 + 14x3 – 4x2 – 13x + 10) + (-8x5 – 2x4 – 7x3 + 3x2 – 3x)

4)  (4x2 – 12x + 29) – (-x + 14x2 + 20)

5)  (8 + 2x2 + 5x3 – 7x) – (11x + 8 – 6x3 – 2x2)

 

(Scroll down for answers)

 

 

 

 

 

 

 

 

 

 



Answers:

1)  11x2 + 11x + 43

2)  8x3 + 3x2 + 5x + 26

3)   -2x4 + 7x3 – x2 + -16x + 10

4)  -10x2 – 11x + 9

5)  11x3 + 4x2 – 18x



 

 

 

Related Links:

Didn't find what you were looking for in this lesson?  More information on polynomials can be found at the following places:


Resource Page


Related Lessons

 

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