Name____________________________________
Date________________ Period______
Advanced Algebra
Closure Property of Polynomials
Previously, you learned how to use manipulatives to add and subtract like terms of polynomial expressions. Now, in this
task, you will continue to use strategies that you previously developed to simplify polynomial expressions. To simplify
expressions and solve problems, you learned that we sometimes need to perform operations with polynomials. We will
further explore addition and subtraction in this task.
Answer the following questions and justify your reasoning for each solution.
1. Bob owns a small music store. He keeps inventory on his xylophones by using x2 to represent his professional
grade xylophones, x to represent xylophones he sells for recreational use, and constants to represent the
number of xylophone instruction manuals he keeps in stock. If the polynomial 5x2 +2x + 4 represents what he
has on display in his shop and the polynomial 3x2 + 6x + 1 represents what he has stocked in the back of his shop,
what is the polynomial expression that represents the entire inventory he currently has in stock?
2. Suppose a band director makes an order for 6 professional grade xylophones, 13 recreational xylophones and 5
instruction manuals. What polynomial expression would represent Bob’s inventory after he processes this
order? Explain the meaning of each term.
3. Find the sum or difference of the following using a strategy you acquired in the previous lesson:
a.
5𝑥 2 + 2𝑥 − 8
+ 3𝑥 2 − 7𝑥 − 1
2𝑥 2 − 2𝑥 + 7
b. (−) 𝑥 2 + 2𝑥 + 1
c. (7x – 5) + (2x + 8)
d. (2a2 – 5a + 1) + (a2 + 3a)
e (-2x2 – 5x + 9) – (-3x2 + 2x + 4)
f. (5x2 + 2xy – 7y2) – (2x2 – 5xy + 2y2)
Name____________________________________
Date________________ Period______
Advanced Algebra
We’ve Got to Operate Learning Task
4. You have multiplied polynomials previously, but may not have been aware of it. When you utilized the
distributive property, you were just multiplying a polynomial by a monomial. In multiplication of polynomials,
the central idea is the distributive property.
a. An important connection between arithmetic with integers and arithmetic with polynomials can be seen
by considering whole numbers in base ten to be polynomials in the base b = 10. Compare the product
213 × 47 with the product (2b2 +1b+3) (4b + 7):
2b 2  1b  3
 4b  7
200  10  3
213
40  7
 47
14b  7b  21
1491
8b 3  4b 2  12b  0
1400  70  21
8000  400  120  0
8b 3  18b 2  19b  21
8000  1800  190  21
2
8520
10011
b. Now compare the product of 135 × 24 with the product (1b2 + 3b + 5) (2b + 4) using the
methods in part a.
5. Find the following products. Be sure to simplify results.
b.  2 x 2 (5 x 2  x  4)
a. 3x(2 x 2  8 x  9)
c.
 2x  7 2x  5

d. (4 x  7)(3x  2)

e.  x  3 2 x 2  3 x  1
g.
 4x  7 y  4x  7 y 
i. ( x  1) 3
f. (6 x  4)( x 2  3x  2)
h.  3 x  4 
j. ( x  1) 4
2
Name____________________________________
Advanced Algebra
Date________________ Period______
We’ve Got to Operate Learning Task
6. A set has the closure property under a particular operation if the result of the operation is always an
element in the set. If a set has the closure property under a particular operation, then we say that the
set is “closed under the operation.”
It is much easier to understand a property by looking at examples than it is by simply talking about it in
an abstract way, so let's move on to looking at examples so that you can see exactly what we are
talking about when we say that a set has the closure property.
a. The set of integers is closed under the operation of addition because the sum of any two integers is
always another integer and is therefore in the set of integers. Write a few examples to illustrate
this concept:
b. The set of integers is not closed under the operation of division because when you divide one
integer by another, you don’t always get another integer as the answer. Write an example to
illustrate this concept:
c. Go back and look at all of your answers to problem number 3, in which you added and subtracted
polynomials. Do you think that polynomial addition and subtraction is closed? Why or why not?
d. Now, go back and look at all of your answers to problems 4 and 5, in which you multiplied
polynomials. Do you think that polynomial multiplication is closed? Why or why not?