Problems for Accelerated Math 2 Name______
Polynomials Period____Date______
1. If p(2/3) = 0, then what is one of the factors of p(x)?
2. What is the remainder when x17 + 13 is divided by x - 1?
3. One of the roots of the function f(x) = 5x2 + k×x - 4 is x = -2. What is the other
root?
4. If Q(x) = ax2 + b×x + c and Q(1) = 9, Q(2) = 21, and Q(3) = 43, find the values of a, b,
and c and use that to write Q(x).
5. If C(x) = ax3 + bx2 + c×x + d and C(-1) = 8, C(0) = -3, C(1) = -4, C(2) = 17, find the
values of a, b, c and d and use that to write C(x).
6. Sketch f(x) = -4(x - 3)(x + 2)
7. Classify each of the following polynomials as even, odd, or neither.
A) f(x) = 4x3 + 8x B) g(x) = x2 + x
C) h(x) = 7x8 + 5x4 + 3 D) k(x) = 3x + 9
8. Write each of the following in both factored and expanded for f(x):
A) A quadratic with roots at 3 and -1 if the leading coefficient is 2.
B) A quadratic with roots at if the leading coefficient is -1.
C) A quadratic with roots at 3i and -3i, if f(2) = 8.
D) A quadratic with leading coefficient of 1, if the sum of the roots is 5 and the
product of the roots is 6.
E) A quadratic with leading coefficient of -3, if the sum of the roots is 2/3 and
the product of the roots is 4.
F) A cubic with roots at 2, 3, and -1, if the leading coefficient is 3.
G) A cubic with roots at 4 and , if the leading coefficient is 2.
H) A cubic with roots at 4 and , if f(1) = 8.
I) A cubic with leading coefficient of 1, if the sum of the roots is 3, the
difference between the smallest and the largest roots is 8, and the sum of
twice the largest and the middle one is 11.
J) A cubic with roots of 2 + i, 2 - i and 1.
K) A cubic polynomial with integral coefficients, p(3) = 0, p() = 0, and
p(0) = -45.
L) A cubic with one root at 5 and a repeated root at 4.
M) A quartic with roots at -3, -1, 1, and 3.
N) A quartic with one root at 1 and a root at -1 with multiplicity of 3.
O) An odd quintic function with at least 3 real roots.
9. Find all the roots of each of the following:
A) f(x) = 2x2 + 3x - 5 B) F(x) = 4x2 + x - 5
C) f(x) = x2 + 4x + 6 D) f(x) = x3 + 3x2 - 6x - 8
E) f(x) = x3 + 4x2 + x - 6 F) G(x) = x3 - 4x2 + 2x + 4
G) P(x) = x3 - 5x2 + 5x + 3 H) f(x) = x3 - 6x2 + 13x -10
I) f(x) = x3 + x2 - x + 15 J) h(x) = 2x3 - 7x2 + 2x + 3
K) P(x) = 3x3 + 2x2 - 19x + 6 L) f(x) = 6x3 - 31x2 + 3x + 10
M) f(x) = x4 -7x3 - 3x2 + 19x + 14 N) p(x) = 2x3 + 4x2 - 17x - 39
O) G(x) = -2x3 - 3x2 + 8x + 12 P) f(x) = x4 + x3 - 7x2 - x + 6
Q) f(x) = x4 + x3 - 7x2 - x + 6 R) P(x) = x4 - 8x3 + 15x2 - 24x + 36
S) H(x) = x4 - 5x3 - 4x2 + 23x - 15 T) f(x) = x4 + x3 - 6x2 - 14x - 12
10. According to Descartes Rule of Signs, how many positive and negative roots will
each of the following polynomials have? Use a calculator to verify this prediction.
Then find all the roots.
A) p(x) = 6x5 + 7x4 - 81x3 - x2 + 189x - 60
B) P(x) = 6x5 + 7x4 - 45x3 + 41x2 - 189x + 60
11. A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. Determine the dimensions that will produce the maximum enclosed area.
y
x x
12. A textile manufacturer has a daily production cost of:
C(n) = 10000 - 110n + .45n2 , where C is the total cost (in dollars) and n is the
number of units produced that day. How many units should be manufactured in
order to minimize the production cost, and what will that cost be?
13. The path of a diver is: , where x is his horizontal distance (in
feet) from the end of the diving board and y is his height (in feet) above the water.
A) What is the maximum height of the dive?
B) When he entered the water, what was his horizontal distance from the board?
14. An open box with locking tabs is to be made from a square piece of paper 24 inches
on a side. This is done by cutting a square with side x from each corner and then
folding along the dotted lines (as shown). x
A) Find the volume of this box as a function of x.
B) Determine the logical domain of this function.
C) If the volume of the box needs to be at least
640 cubic inches, what should x be?