Math 3
3-8 Multiplication with Polynomials
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Name_______________________________
I can perform polynomial multiplication
DEMAND:
Think about how much money you’d be willing to pay for your favorite pair of shoes, to see your favorite
musical artist in concert, or even a good burrito. Would you pay $1, $2, $5, $10, $50, $100, $500,
$1000? As the price went up, what happened to your desire to pay that amount? In any business,
raising the price will lower the demand for that product. Business owners want to sell their product for
as high of a price as possible (to maximize their incomes and their profit) without scaring away too many
customers. Remember, if they set the price for their product too high, not enough people will want the
product and they won’t make a profit.
Based on our work in the last section with adding polynomials, and our new understanding of how
business expenses work, we decide that this owning pizzeria thing sounds pretty good. We decide to
start our own pizzeria. After some careful analysis, we figure out that it will cost us $2 to make a quality
pizza and our daily operating expenses will be $800.
Write a function, E ( q ) , where E ( q ) is the daily operating expenses for our pizzeria and q is the
quantity of pizzas made. Remember: EXPENSES = VARIABLE COSTS + FIXED COSTS.
E ( q)  __________________
For our pizza business, let’s assume that, even if we give away our pizzas for free, the demand for our
pizzas would max out at 300 pizzas per day. With some business analysis, we discover that for every
dollar we raise the price of our pizza, it lowers the demand (or number of pizzas we would sell) by an
average of 7 ½ pizzas per dollar. Therefore, our demand function would be q( p )  7.5 p  300 ,
where q is the quantity demanded and p is the price of our pizza.
We are still interested is seeing what our total expenses will be, but they will vary based off the demand
for our pizzas. Since the demand for our pizzas is driven by the price we sell our pizzas for, our expenses
are related to the price of our pizza.
If E (q )  2q  800 and q( p )  7.5 p  300 , let’s use the substitution method to find a new function,
E ( p) , where our business expenses are based off of the price we sell our pizza for.
Substitute 7.5 p  300 in for q in E ( q ) .
E (q)  2q  800
q( p )  7.5 p  300
E ( p )  ____________________
NOTICE: E ( q ) relates our Expenses to the quantity of pizzas made. E ( p ) relates our Expenses to
the price we choose to sell our pizzas for (which will affect the demand for our pizzas).
Because we substituted q ( p ) into E ( q ), another way of representing E ( p ) is E ( q( p )) . Although
this notation may look strange, it makes sense. With parentheses, we work our way inside out, meaning
we start with the inside, q ( p ) , and plug into the outside, the E function.
Let’s apply this same process to Bennedetto’s and Cala’s.
Bennedetto’s had an expense function of B( q)  2.65q  1100 . They know that the demand for their
pizza is modeled by the function q( p )  8 p  400 . Substitute q ( p ) into B ( q ) to find the daily
expenses for Bennedetto’s, based off of the price of their pizza.
B( p )  ______________________
Cala’s had an expense function of C ( q)  2.20q  740 . They know that the demand for their pizza is
modeled by the function q( p )  6.8 p  306 . Substitute q ( p ) into C ( q ) to find the daily expenses
for Cala’s, based off of the price of their pizza.
C ( q( p ))  C ( p )  ______________________
What will our pizzeria expenses be if we set our pizza price at $5 per pizza? Evaluate E (5).
E (5)  __________
Evaluate the following:
E (9)  __________
E (12)  __________
B(10)  __________
C (8)  __________
REVENUE OR INCOME:
Businesses sell their product or services in order to make money. The money that businesses take in
from their customers is called income or revenue. Businesses want as much revenue as possible
without incurring too much cost. Remember, it takes money to make money, but in the end, your costs
are deducted from your income.
If we sell 100 pizzas for $1, what would our income be? ___________
If we sell 90 pizzas for $2, what would our income be? ___________
If we sell 50 pizzas for $10, what would our income be? ___________
If we sell 5 pizzas for $20, what would our income be? ___________
In each of the above scenarios, how did you calculate our income?
What happened to our income as we raised the price of our pizzas?
No matter what the business is, REVENUE = PRICE x QUANTITY SOLD
In other words, the price of our pizza, p , is multiplied by our demand function, q( p )  7.5 p  300 .
Therefore, for our Revenue function, we multiply p  (7.5 p  300) .
Distribute to simplify R( p)  p  (7.5 p  300) .
R( p )  ______________________
Do the same to calculate the Revenue function for Bennedetto’s and Cala’s.
RB ( p)  p  ( 8 p  400)
RB ( p)  ______________________
RC ( p)  p  ( 6.8 p  306)
RC ( p)  ______________________
Note: RB ( p ) stands for the revenue at Bennedetto’s based off of the price per pizza. Similarly, RC ( p)
represents the revenue at Cala’s based off of the price per pizza.
Let’s say that things are going well at our pizzeria, and we are considering increasing the price of our
pizza by $1. This will change our revenue model. Let’s recalculate our revenue to reflect our price
change.
R( p )  ( p  1)  ( 7.5 p  300)
R( p )  ________________________________
Write the polynomial in standard form
Bennedetto’s is considering raising prices by $2. This will change their revenue model. Let’s recalculate
their revenue to reflect the price change.
RB ( p)  ( p  2)  ( 8 p  400)
RB ( p)  ________________________________
Write the polynomial in standard form
Cala’s is considering lowering prices by $1. This will change their revenue model. Let’s recalculate their
revenue to reflect our price change.
RC ( p)  ( p  1)  ( 6.8 p  306)
RC ( p)  ________________________________
Write the polynomial in standard form
The process above is called polynomial multiplication.
Practice:
Use the following functions to perform polynomial function multiplication. Write your answers in
standard form.
f ( x )  3x  2
g ( x )  2 x  5
h( x )  x 2  4 x  7
j ( x )  3x 2  x  8
Multiply. Write your answers in standard form.
f ( x )  h( x )
g ( x)  j( x)
[ f  h ]( x )  ____________________________ [ g  j ]( x )  _____________________________
h( x )  j( x )
[h  j ]( x )  ________________________________________
Evaluate. Use the functions and your answers from above.
f (5)  ________
h(5)  _________
Now multiply your previous two answers.
f (5)  h(5)  _________
Now try evaluating your function [ f  h ](5) . That is, use your function from the previous page,
[ f  h ]( x ) , and evaluate the function substituting 5 for x.
[ f  h ](5)  _________
Did you get the same result for f (5)  h(5) and [ f  h ](5) ? Which method of evaluating was
faster?