POLYNOMIAL FUNCTIONS
75
POLYNOMIAL FUNCTIONS
A
polynomial function
f (x) has a general equation f (x) = a
1
x
n
+ a
2
x
n−1
+a
3
x
n−2
+ . . . + a
z
, where coefficients and constants are associated with
a and
exponents associated with
n are positive integers. Linear functions, such as
y = 3x − 5, and quadratic functions, such as r = 3w
2
− 5w + 7, are polynomial
functions that have numerous applications discussed elsewhere in this book (see
Linear Functions and
Quadratic Functions). Polynomial functions with degree
three or greater are found in applications associated with volume and financial
planning.
Empty open-faced containers such as crates are put together by attaching a
net of five rectangles. A rectangular piece of plastic can be cut so that it can turn
into an open-faced rectangular prism when folded at its seams. If a square piece
is cut out of each corner of a rectangle, then four folds will form a net with five
rectangles that can
be formed to develop the prism, as shown below.
A manufacturer is probably interested in finding the location to cut the square
from the corners so that the consumers will be able to fill the crate with the most
amount of material. In essence, the goal is to maximize the volume based on a
fixed amount of material. Suppose that square corners are removed from a rec-
tangular sheet of plastic with dimensions of 6 feet by 4 feet. Each side of the
prism can be represented in terms of the edge length,
x,
of the square that was
removed from the corners, as shown below.
The volume of the crate,
v, is the product of its dimensions, so it can be rep-
resented by the equation
v = x(6 − 2x)(4 − 2x). This equation is a polynomial
function, because it is the factored form of
v = 4x
3
− 20x
2
+ 24x.
A relative
maximum of this function on a graph, as shown on the following page, within a
domain between 0 and 2 feet occurs when
x ≈ 0.78 feet, or about 9.4 inches.
This means that the crate with the largest possible volume will occur when
squares with an edge length of 9.4 inches are cut from the corners.
Open-faced prism with dimensions
x
by l − 2x
by w − 2x
formed by cutting squares
with side length
x
out of the corners of rectangular sheet with dimensions l
by w
.
Open-faced prism formed by cutting
squares with edge length, x, out of the
corners of a rectangular sheet with
dimensions of 6 feet by 4 feet.
Long-term investing uses a polynomial function to account for money that is
invested each year. Suppose an account was set up so that you contributed money
each year towards your retirement based on
a fixed percentage of interest, assum-
ing that you continued to add a minimum amount of money to the account each
year and did not withdraw money at any time. The total amount of money,
m, in
the bank after
n years based on an annual interest rate of r percent can be repre-
sented by the function
m = a
1
(1 +
r
100
)
n
+a
2
(1 +
r
100
)
n−1
+a
3
(1 +
r
100
)
n−2
+ . . . + a
z
,
where the coefficients,
a, are the individual amounts
of money deposited into the
account after each year. For example, if $500 is deposited at the end of the first
year, $700 at the end of the second year, $800 at the end of the third year, and
$400 at the end of the fourth year, then the total amount of money in the account
at the end of the fourth year is determined by the equation
m = 500(1 +
r
100
)
3
+ 700(1 +
r
100
)
2
+ 800(1 +
r
100
) + 400.
This means that the initial deposit of $500
will compound three times, the second
deposit of $700 will compound two times, and so on. If an employee uses this
retirement plan for only four years and wants to know the value of the account
21 years after the first investment, then the equation would be rewritten to
m = 500(1 +
r
100
)
20
+ 700(1 +
r
100
)
19
+ 800(1 +
r
100
)
19
+ 400(1 +
r
100
)
18
.
This information is useful for people in their financial planning so that they can
learn how to save money for their children’s education and their own retirement.
online sources
for further exploration
Antenna pattern correction
Application of polynomial functions
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