Make sure to do similar problems (such as from the book)
until you are
comfortable with that type of problem!
Answer the following questions (1 -5) based
on the polynomial P(x)
1. Find P(-5) =
5. What are the (x,y) coordinates of the absolute maximum:
6. The length of a rectangular box is 1 inch more than twice the height of the
box, and the width is 3 inches more than the height. If the volume of the
box is 126 cubic inches, find the dimensions of the box.
Answer: height = 3 inches, width = 6 inches, length = 7 inches
7. Let P(x) = x4 + x3 - x2 + x – 2.
a. Find all the zeros of P(x).
Answer: 1, -2, i, -i
b. Completely factor P(x) into a product of linear terms.
a. Use either long division or synthetic division to show that (x+2) is a
linear factor of P(x).
b. Show that (x+2) is a linear factor of P(x) without using division (hint:
it’s a Theorem).
Answer: P(-2) = 0 (factor thm P.317).
c. Use your results to rewrite P(x) as a product of two factors, one linear
and one non-linear:
Answer: P(x) = (x+2) (x2 +2x -15)
9. Given has a zero "-3" with multiplicity two.
Find the other zeros (real or imaginary). Show your work.
Answer: 3i and -3i.
10. Write a third degree polynomial with zeros 4i and 2 in standard form.
One possible Answer: .
Explanation (nov. 24, 5:30pm) Since 4i and 2 are both roots of the polynomial,
x-4i and x-2 are both factors of the polynomial, this only gives us a second
degree polynomial though. I
11. The graph of P(x) is shown below.
a. Write a polynomial in factored form
that has the same general shape and
the same zeros.
One possible answer:
b. What is the degree of P(x)? _ 4 ___
12. Given :
a. What is the degree of P(x)? __ 5 __
b. What are the zeros of P(x)? __ -2 and 5 _______
c. What is the multiplicity of each zero? -2 has mult. 4 and 5 has mult. 1
d. What is the end behavior of P(x)? up to the left and down to the right
e. Where does P(x) cross the x-axis, if at all? __ at x=5 _____
f. Where does P(x) bounce off the x-axis, if at all? _ at x = -2 ___
13. For each of the functions below determine the horizontal and vertical
asymptotes. If none exist, then write “none”.
|x-axis||y = 3||None|
|x = -3 and x = 5||x = -1/2||y-axis|
Graph f(x): Use a dotted line to indicate
Answer: horizontal asymptote at y=1, vertical
symptote at x=3 (graph on
calculator to check – make sure you label key points!)
14. Find the exact polynomial that matches the
graph exactly, i.e. it passes through the
point (-4,12) and has the same zeros.
Explanation (nov. 24, 6pm) Since the real roots
are -3, 1, and 4, the polynomial has the factors
(x+3), (x-1), and (x-4). Since it bounces off at -3,
the root -3 has even multiplicity (p. 331). We are
told that the equation of the polynomial should
have the exact same zeros as the graph. Also,
this graph has the shape of a 4th degree polynomial. Since -3
multiplicity, and we have 3 roots so far, 3 must have multiplicity 2. Therefore, our
polynomial has the factors (x+3)2, (x-1), and (x-4). We are almost there… we
cannot quite conclude that our polynomial is (x+3)2(x-1)(x-4) because we need to
make sure it goes through the point (-4,12). So our polynomial may have a
leading coefficient of something other than 1. That is our polynomial is of the
form p(x)=c(x+3)2(x-1)(x-4), where c is a constant to be determined. Since it
goes through the point (-4,12), we know that p(-4)=12. Plugging in -4 for x and
setting it all equal to 12 then solving for c gives that c =3/10. So
p(x)=(3/10)*(x+3)2(x-1)(x-4) and multiplying out these factors gives the answer
15. Match each function with its graph. Record the letter
of the graph next to the
16. Complete the table below by changing the form of the equation.
|Exponential Form||Logarithmic Form|
|Ans: log (1) = x|
17. Write as a single logarithm with a coefficient of 1. Box answer.
18. Write the logarithm in terms of logarithms x, y, and z. Box answer.
19. Name that inverse! (Find the inverse of each function.)
20. Calculate without