5.2PolynomialOperations
At times we’ll need to perform operations with polynomials. At this level we’ll just be adding, subtracting,
or multiplying polynomials. Dividing polynomials will happen in future courses. Oh happy day. ☺
Adding/Subtracting Polynomials
Recall that addition means “combining like things” and that subtraction is really just “adding the opposite.”
These two definitions should carry us through these operations. If addition means “combining like things,” then
when we add polynomials we can only add the like terms. The quadratic terms are alike (they’re all to the second
power), so we can add all the quadratic terms. The cubic terms are alike (they’re all to the third power), so we can
add all the cubic terms. I could type sentences like that until I’m blue in the face, or we could just look some
examples. Let’s just look at some examples.
2 C − 4 + − 9) + ( − 7 − 4)
In this case, we might look for all the cubic terms first. Notice that there is no cubic term in the second
polynomial, so the 2 C will stay 2 C . Now let’s look for quadratic terms.
(2 C − 4 + − 9) + ( − 7 − 4)
We have a −4 and an to add. Since the coefficient on the is really the understood 1, we’ll end up
with −3 . In a similar fashion, we can combine the linear terms and the constant terms. So we might mark all the
terms that can be combined as follows to get our final sum.
2 C − 4 + − 9 + − 7 − 4
2 C − 3 − 6 − 13
Subtraction works exactly the same way. Just remember to distribute the subtraction sign through the
second polynomial. Since subtract means “add the opposite,” we have to take the opposite of every term in the
second polynomial.
4 C − 7 − 6 − 3 C − 4 − 7 − 8
C + 4 + 2
Notice that there was no quadratic term in the first polynomial, so we just subtracted the negative quadratic
from the second polynomial. This gave us the +4 term. Also note that the linear terms became zero (you might
think of “canceling”) because we had −7 − −7 which is really −7 + 7.
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Multiplying Polynomials
Multiplying polynomials really boils down to the distributive property. Let’s step through an example of a
monomial times a polynomial, and then we’ll step it up a bit.
2 C ( − 3 + 4)
2 D − 6 + 8 C
Now let’s look at a binomial times a trinomial. In this case we can think of the binomial as a single entity
that we are distributing to every term of the trinomial as follows.
(2 − 1 + 3 − 4
2 − 1 + 32 − 1 − 42 − 1
Since we are still not completely simplified, we’ll need to distribute again as follows.
2 − 1 + 32 − 1 − 42 − 1
2 C − + 6 − 3 − 8 + 4
Finally, combine like terms to simplify.
2 C + 5 − 11 + 4
The question now becomes whether or not there is an easier way to do this multiplication than the double
distributive property. Notice that each term in the binomial 2 − 1 got multiplied by each term in the trinomial
+ 3 − 4. In other words, the 2 got multiplied by each term in the trinomial. Then the −1 got multiplied by
every term in the trinomial. This shows us that really the distributive property can be expanded to mean “each
term in the first parentheses times each term in the second parentheses.”
2 − 3 − 2 + 5 + 3
2 ∗ + 2 ∗ 5 + 2 ∗ 3 − 3 ∗ − 3 ∗ 5 − 3 ∗ 3 − 2 ∗ − 2 ∗ 5 − 2 ∗ 3
2 + 10 C + 6 − 3 C − 15 − 9 − 2 − 10 − 6
2 + 7 C − 11 − 19 − 6
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A Closed System
Alright, let’s think about the integers. Now, you may be thinking, “Why are we switching topics to the
integers?” Just stick with me. We’ll get there.
So, the integers. One of the crazy things about the integers is that they form what is called a closed system
under the operations of addition, subtraction, and multiplication. A closed system means that if you add two
integers, you get an integer answer. If you subtract two integers, you get an integer answers. If you multiply two
integers, you get an integer answer. A closed system means if you perform operations, what you start with is what
you end up with.
You should probably be able to see why the integers aren’t closed under the operation of division. While
some integers divided by integers will give you integer answers (such as four divided by two), most give you a
rational answer (like five divided by two).
Rabbits also have a closed system under addition, subtraction, and multiplication. If you add rabbits and
rabbits, you get rabbits. If you subtract rabbits from rabbits, you get rabbits. If rabbits multiply, you get more
rabbits. That is a closed system.
Now take a guess as to whether or not polynomials are a closed system. If you add two polynomials, do
you get another polynomial? If you subtract two polynomials, do you get another polynomial? If you multiply two
polynomials, do you get another polynomial? Take a look at the examples on the previous page and you should see
that the polynomials are closed over addition, subtraction, and multiplication.
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Lesson 5.2
Perform the following polynomial operations.
1. 1 + C + 2 + 8
2. 3 − 4 − + 3 + 5
3. 2 − 1 C + 4 + 3
4. + 3 − 5 − + 4
5. 4 C + 6 − 2 + 2 − 4
6. D − C + 4 − 4 C + 2
7. 2 C + 24 − 3 − 2
8. + 2 − 52 − 3 − 3
9. 6 − 1 + C + − + 1
10. − + 7 − D + 4 11. 2 C 4 − 2 − 1
12. 3 C − 4 − 2
13. 7 D + + 2 + + 3 D 14. 5 − 4 − 6 − 2 + 15. 6 − + 3 + 4
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16. + 13 + 2 + 1
17. 5 + 3 + C + − 6 + 3
18. 2 D + C − 4 − − 4 C 19. 3 − 9 + 1 + − 2
20. 4 + 3 − 7 − 2
21. 5 + D + 2 D + C − 22. 4 − − + 3 + 4
23. 2 − 39 + 4 C + 2 D 24. 2 + C + 3 + 4
25. What degree of polynomial would you get if you added a 5th degree polynomial to a 3rd degree polynomial
and how do you know?
26. If you added a trinomial to a binomial, how many terms could the sum have?
27. If you add or subtract two polynomials, why do you always get another polynomial?
28. If you multiply two polynomials, why do you always get another polynomial?
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