Name____________________________ Period______ Date_________
Algebra of Functions – Activity 2
Addition, Subtraction, and Multiplication of Polynomial Functions
Multiplication of Binomials
A procedure for multiplying binomials is called the FOIL method. FOIL is an acronym for the sum of the
products of the first, outer, inner, and last terms of the binomials. Essentially, you multiply each term of
the first binomial by each term of the second binomial.
1. a. Given 𝑓(𝑥) = 𝑥 + 7 and 𝑔(𝑥) = 𝑥 + 5, determine a single polynomial expression for 𝑓(𝑥) ∙ 𝑔(𝑥)
by multiplying (𝑥 + 7)(𝑥 + 5). Write your answer as a sum of terms.
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b. Using the functions defined in part a, complete the following table.
𝑥
𝑓 (𝑥 )
𝑔(𝑥 )
𝑓(𝑥) ∙ 𝑔(𝑥)
0
1
2
3
4
c.
When you determine the product function, what do you multiply: domain values, range
values, or both? Explain.
Multiply Powers Having the Same Base
The volume, 𝑣 (in cubic feet), of a partially cylindrical storage tank of liquid fertilizer is represented by
the formula 𝑣(𝑟) = 𝑟 2 (4.2𝑟 + 37.7) where 𝑟 is the radius (in feet) of the cylindrical part of the tank.
2. Determine the volume of the tank if its radius is 3 feet.
Suppose you were asked to write the expression 𝑟 2 (4.2𝑟 + 37.7) as an equivalent expression without
parentheses. Using the distributive property, you would multiply each term within the parentheses by
𝑟 2 . The first product is 𝑟 2 (4.2𝑟). What is 𝑟 2 times 𝑟?
Recall that in the expression 𝑟 2 the exponent 2 tells you that the base 𝑟 is used as a factor two times. In
the expression 𝑟, the exponent, 1, tells you that the base 𝑟 is used as a factor once.
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3. a. Complete the following table.
𝐼𝑁𝑃𝑈𝑇 𝑟
𝑂𝑈𝑇𝑃𝑈𝑇 𝑓𝑜𝑟 𝑟 2 ∙ 𝑟
𝑂𝑈𝑇𝑃𝑈𝑇 𝑓𝑜𝑟 𝑟 3
2
4
5
b.
How
does the table
demonstrate that 𝑟 2 ∙ 𝑟 is equivalent to 𝑟 3 ?
c. Consider the following products.
What pattern do you observe?
Multiplication Property of Exponents
Let 𝑚 and 𝑛 be rational numbers.
To multiply powers of the same base, keep the base and add the exponents.
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
4. Expressions for 𝑝(𝑥) and 𝑞(𝑥) are given in the following table. Fill in the last column of the table
with a single power of 𝑥.
𝑝 (𝑥 )
𝑥2
2𝑥 3
−3𝑥 5
5𝑥 4
𝑞 (𝑥 )
𝑥4
𝑥
4𝑥 2
3𝑥 4
𝑝(𝑥) ∙ 𝑞(𝑥)
5. Multiply (−2𝑎5 )(8𝑏3 )(3𝑎2 𝑏). Explain the steps you used to determine this product.
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6. If 𝑓(𝑥) = 2𝑥 2 + 3 𝑎𝑛𝑑 𝑔(𝑥) = 5𝑥 3 − 2, determine 𝑓(𝑥) ∙ 𝑔(𝑥).
When multiplying polynomials with more than two terms, FOIL cannot be applied. However, the
geometric principles behind the FOIL method can still be applied, or you can simply multiply each term
of the first polynomial by each term of the second and collect like terms.
7. a. Multiply (4𝑥 + 2)(𝑥 2 − 4𝑥 + 3). Determine the appropriate products to complete the chart
below. Combine like terms, and write the final answer for the product in descending order of the
exponents.
𝑥2
−4𝑥
+3
4𝑥
+2
b.
Multiply (4𝑥 + 2)(𝑥 2 − 4𝑥 + 3) by multiplying each term of the polynomial by each term
of the second. Combine like terms, and write the final answer for the product in descending
order of the exponents.
c. How do the final answers in parts a and b compare?
Special Properties
8. Use 𝐹𝑂𝐼𝐿 or the rectangular method to determine the product (𝑥 + 4)(𝑥 − 4)
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Notice that the outer and inner products subtract out. The general form for this product is the following
identity:
(𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2
This expression, 𝑎2 − 𝑏 2 , is the difference of the squares of the binomial terms. For Problem 15, 𝑎 = 𝑥
and 𝑏 = 4 so 𝑎2 = 𝑥 2 and 𝑏 2 = 42 = 16. Then 𝑎2 − 𝑏 2 = 𝑥 2 − 16.
9. To use the identity (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 to determine the product (2𝑥 + 3)(2𝑥 − 3),
complete the following steps.
a. Identify 𝑎 and 𝑏
b. Identify 𝑎2 and 𝑏 2
c. Write the product as 𝑎2 − 𝑏 2
10. Use FOIL or the rectangle method to determine the product (𝑥 − 4)2 = (𝑥 − 4)(𝑥 − 4)
Notice that the outer and inner products are equal. The general form for this product is the following
identity:
(𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2
This expression, (𝑎 + 𝑏)2 , is the square of a binomial. For Problem 17, 𝑎 = 𝑥 and 𝑏 = −4 so 𝑎2 = 𝑥 2
and 𝑏 2 = (−4)2 = 16. Then (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 = 𝑥 2 + 2 ∙ 𝑥 ∙ (−4) + 42 = 𝑥 2 − 8𝑥 + 16.
11. To use the identity(𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 to determine the square of the binomial
(2𝑥 + 3)2 , complete the following steps.
a. Identify 𝑎 and 𝑏
b. Identify 𝑎2 and 𝑏 2 and 2𝑎𝑏 c. Write the product as 𝑎2 + 2𝑎𝑏 + 𝑏 2
12. Use the identity (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 𝑜𝑟 (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 to determine the
following products.
a. (𝑥 + 6)(𝑥 − 6)
b. (𝑥 − 6)2
b.
(4𝑥 + 1)(4𝑥 − 1)
d. (3𝑥 + 5)2
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Summary
1. To multiply any two polynomials, multiply each term of the first by each term of the second.
2. A common method to multiply two binomials is the FOIL method
First Multiply the FIRST terms in each binomial
Second Multiply the OUTER terms
Third Multiply the INNER terms
Fourth Multiply the LAST terms
Fifth Sum the products in steps 1-4.
You can also use the geometry model.
3. Given two functions, 𝑓 and 𝑔, the product function is defined by 𝑦 = 𝑓(𝑥) ∙ 𝑔(𝑥)
4. To multiply powers of the same base, keep the base and add the exponents. Symbolically, this
product of exponents is written as 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 where 𝑚 and 𝑛 are rational numbers.
5. Special products:
a. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2
b. (𝑎 + 𝑏)(𝑎 + 𝑏) = 𝑎2 + 2𝑎𝑏 + 𝑏 2
Practice
1. a. You are drawing up plans to enlarge your square patio. You want to triple the length of one
side and double the length of the other side. If x represents a side of your square patio, write an
expression for the new area in terms of 𝑥.
b. You discover from the plan that after doubling one side of the patio, you must cut off 3 feet
from that side to clear a shrub. Write an expression in terms of x to represent the length of
this side.
c. Use the result from part b to write an expression without parentheses to represent the new
area of the patio. Remember that the length of the other side of the original square patio
was tripled.
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2. A rectangular bin has the following dimensions.
a. Write an expression that represents the area of the base of the bin.
b.
Using the result from part a, write an expression that represents the volume of the bin.
(Note that the volume is computed by multiplying the area of the base by the height.)
3. You are working for a concert promoter and she has assigned you the task of setting the ticket
prices. She knows from experience that you will sell 3000 tickets if you price them at $40 each. She
also knows that you will sell 100 more tickets for every dollar that you reduce the ticket price. Your
job is to determine the ticket price that will maximize the revenue for the concert.
a. Let 𝑥 represent the number of $1 reductions in the price of the tickets. Write an equation
for the price of each ticket, 𝑃(𝑥).
b. Write an equation for the number of tickets sold, 𝑁(𝑥).
c.
The revenue is the total amount of money collected. In this case the revenue will be
determined by multiplying the number of tickets sold by the cost of each ticket. Determine
an equation for the total revenue, 𝑅(𝑥), as a function of the number of $1 reductions in
price, 𝑥.
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d. What is the domain of the revenue function if the promoter has informed you that she will
not sell the tickets for less than $30 each? (hint – look at underlined section)
e. Complete the following table.
Number of $1
0
Reductions, 𝑥
Price per Ticket,
𝑃(𝑥)($)
Number of Tickets
Sold, 𝑁(𝑥)
Total Revenue,
𝑅(𝑥)($)
f.
2
4
6
8
10
How do the values in the fourth row of the table in part e relate to the values in the second
and third rows?
g. Rewrite the cost function by multiplying the factors and then combining like terms.
h. Calculate the values using your expression from step g for the values 𝑥 = 0, 2, 6.
𝑥
𝐶(𝑥)
0
2
6
i.
How do these values compare to the table?
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4. Use the property of exponents 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 to determine the following products.
a. 35 ∙ 37
b. 𝑡 4 ∙ 𝑡
c. 𝑥 2 ∙ 𝑦 5
d. (2𝑧 4 ) ∙ (3𝑧 8 )
e. (−2𝑥)(3𝑥 4 )(−5𝑥 3 )
f. (𝑎2 𝑏 2 )(𝑎3 𝑏4 )
g. 𝑥 2𝑛 ∙ 𝑥 𝑛
5. Multiply (𝑥 + 3)(𝑥 2 + 3𝑥 − 5). Determine the appropriate products to complete the chart.
Combine like terms, and write the final answer for this multiplication in descending order of the
exponents.
6. Multiply(𝑥 2 + 2𝑥 − 3)(2𝑥 2 + 3𝑥 − 4) . Determine the appropriate products to complete the chart.
Combine like terms, and write the final answer for this multiplication in descending order of the
exponents.
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7. Determine each product, and simplify the results.
a. (3𝑥 + 2)(2𝑥 + 5)
b. (3𝑥 − 2)(2𝑥 − 5)
c. (𝑥 + 2)(4𝑥 − 3)
d. (𝑥 − 2)(4𝑥 + 3)
8. Determine the following products, and simplify the results.
a. (2𝑥 + 5)(𝑥 − 3)
b. (4𝑥 + 3)(3𝑥 − 2)
c. (𝑥 + 2)(𝑥 2 + 4𝑥 − 3)
d. (4 − 3𝑥 + 𝑥 2 )(2𝑥 2 + 𝑥)
e. (𝑥 − 3)(2𝑥 2 − 5𝑥 + 1)
f. (𝑥 − 4)(4 − 𝑥 2 )
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g. (𝑥 2 − 3𝑥 + 1)(3𝑥 2 − 5𝑥 + 2)
h. (2𝑥 2 + 5𝑥)(6 − 2𝑥)
9. a. Expand (2𝑥 + 3)2
b. Multiply (3𝑥 − 2)2
c. Multiply (5𝑥 + 2)(5𝑥 − 2)
d. Multiply (𝑥 2 + 5)(𝑥 2 − 5)
e. After simplifying in parts c and d, the product contains only two terms. Explain why. (Hint:
Compare the first terms to each other and the second terms to each other.)
10. a. Given 𝑓(𝑥) = 𝑥 + 1 and 𝑔(𝑥) = 2𝑥 − 3, determine 𝑓(𝑥) ∙ 𝑔(𝑥) by multiplying and combining
like terms.
b. Use 𝑓 and 𝑔 as defined in part a to complete the following table.
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𝑥
𝑓(𝑥 )
𝑔(𝑥 )
= 𝑥+1
= 2𝑥 − 3
𝑓(𝑥) ∙ 𝑔(𝑥)
Function from
10a:
0
1
2
3
4
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