Name _______________________
Date ______________ Hour ____
Answer each of the following questions. Show all of your work.
1. Simplify each expression.
a) (3𝑥 2 − 4𝑥 + 7) + 2(𝑥 2 − 8)
b) (𝑥 2 − 6𝑥 + 1) − (5𝑥 3 − 2𝑥 2 − 8)
c) (2𝑥 2 + 6𝑥)(−𝑥 2 + 2𝑥 − 5)
d) (6𝑥 4 − 3𝑥 2 + 𝑥) − (2𝑥 3 + 5𝑥 2 + 4𝑥) + (𝑥 2 − 𝑥)
e) (2𝑥 2 − 𝑥 + 3)(𝑥 2 + 4𝑥 + 5)
2. Write the polynomial in standard form.
a) 𝑓(𝑥) = 2𝑥(𝑥 − 4)2
b) 𝑝(𝑥) = (3 − 𝑥)(𝑥 + 2)(4𝑥 − 1)
3. Divide using long division. Write your final answer as a polynomial.
a) (4𝑥 4 + 𝑥 3 − 4𝑥 2 + 7𝑥 + 2) ÷ (4𝑥 + 1) b) (3𝑥 4 + 15𝑥 3 − 29𝑥 2 − 25𝑥 + 40) ÷ (3𝑥 2 − 5)
4. Divide using synthetic division. Write your final answer as a polynomial.
a) (6𝑥 3 − 11𝑥 2 − 51𝑥 + 56) ÷ (2𝑥 − 7)
b) (𝑥 4 + 𝑥 3 – 1) ÷ (𝑥 − 2)
5. Use the Remainder Theorem and synthetic division to find P(c).
a) 𝑃(𝑥) = 3𝑥 3 + 𝑥 2 + 𝑥 − 5, c = 2
b) 𝑃(𝑥) = 𝑥 4 − 10𝑥 3 + 2, c = 3
6. Use the Factor Theorem to show the given binomial is a factor. Then, factor the polynomial
completely and find all of the zeros.
a) 𝑝(𝑥) = 𝑥 3 − 8; 𝑥 − 2
b) 𝑝(𝑥) = 25𝑥 3 + 110𝑥 2 + 140𝑥 + 40; 5𝑥 + 2
c) 𝑝(𝑥) = 3𝑥 4 − 14𝑥 3 + 27𝑥 2 − 56𝑥 + 60; 3𝑥 − 5
7. What is a root to a polynomial? Is there a way to tell how many solutions a polynomial should
have? How do you know?
8. Determine whether each number is a root of 𝑎3 − 3𝑎2 − 3𝑎 − 4 = 0. Explain.
a) 0
b) 4
c) -2
9. Is x = 2 a solution to the polynomial 𝑓(𝑥) = 4𝑥 3 − 5𝑥 2 + 3𝑥 − 18? Explain.
10. Write a polynomial equation of least degree with roots of -1, 1 and 5.
11. Write a fourth degree polynomial with zeros of 2, -4, and 0 (multiplicity 2).
12. What are complex conjugates? Give an example.
13. Write a third degree polynomial equation with the zeros −3𝑖 and 6.
14. Write a fourth degree polynomial with zeros 2 − 𝑖 and 5𝑖.
15. Write a third degree polynomial with zeros 3 + 4𝑖 and −6.
16. Find the roots of each polynomial (factor):
a) 4𝑥 3 − 12𝑥 2 + 𝑥 − 3 = 0
b) 2𝑥 4 − 2𝑥 2 − 24 = 0
c) 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 + 4𝑥 − 8
d) 𝑓(𝑥) = 3𝑥 3 − 𝑥 2 − 27𝑥 + 9
17. Factor each polynomial function completely. Then, find all of the zeros.
a) 𝑥 3 + 5𝑥 2 − 9𝑥 − 45 = 0
b) 2𝑥 3 + 7𝑥 2 + 18𝑥 + 63 = 0
18. A rectangular box has a volume of 𝑉(𝑥) = 𝑥 3 + 10𝑥 2 + 31𝑥 + 30 cubic inches. The height of the
box is x + 2 inches. The width of the box is x + 3 inches. Find the length of the box in terms of x.
19. Mrs. Koenig bought an art studio which has the floor plan
like the diagram below. She needs to order tile, but she
needs to know the dimensions of the room before she can
call the hardware store. Write a polynomial function, in
standard form and in terms of x, for the area of the art
studio. Then, find the value of x if the studio is to be 355
square feet.
x+1
x
x+2
x
2x + 3
2x
20. The floor space in square feet of retail stores A, B, and C can be modeled by 𝐴 = 𝑥 2 + 4𝑥 − 7, 𝐵 =
2𝑥 2 − 7𝑥 + 1, and 𝐶 = −4𝑥 2 + 250𝑥 − 1. Write a model for the total amount of floor space T for
all three stores.
21. Suppose you have 250 cubic inches of clay with which to make a sculpture shaped as rectangular
prism. You want the height and width each to be 5 inches less than the length. What should the
dimensions of the prism be?
22. A wooden board is shaped like a rectangular prism. It has a total volume of 324 cubic inches. The
width is 3 inches less than the height and the length is 12 inches longer than the height. What are
the dimensions of the board?
23. You have a rectangular piece of steel whose dimension are 20 inches by 16 inches. You are
required to cut out the four corners of the rectangle so that you may fold up the side to create a
box. Write a function you would use to find the volume of the box if x represents the length of the
cuts.
24. From 1991 through 1998, the number of commercial C and education E Internet websites can be
modeled by the following equations, where t is the number of years since 1991.
𝐶 = 0.321𝑡 2 − 1.036𝑡 + 0.698
𝐸 = 0.099𝑡 2 + 20.12𝑡 + 0.295
Find a model for the total number of commercial and education sites.