Question

1.A mysterious box is delivered to the dinner party you are attending. The label on the box says that the volume of a box is the function f(x) = x3 + 3x2 – 10x – 24. To open the box, you need to identify the correct factors of f(x). Party-goers offer up solutions, and it is your job to find the right ones.


Their suggestions are:
•(x – 1)
•(x + 2)
•(x – 3)
•(x + 4)
•(x + 6)
•(x – 12)


List the correct factors. Then justify your selections with complete sentences.

2.Three party-goers are in the corner of the ballroom having an intense argument. You walk over to settle the debate. They are discussing a function g(x). You take out your notepad and jot down their statements.

•Professor McCoy: She says that 2 is a zero of g(x) because long division with (x + 2) results in a remainder of 0.
•Ms. Guerra: She says that 2 is a zero of g(x) because g(2) = 0.
•Mr. Romano: He says that 2 is a zero of g(x) because synthetic division with 2 results in a remainder of 0.


Correct the reasoning of any inaccurate reasoning by the party-goers in full and complete sentences. Make sure you reference any theorems that support your justifications.

3.Dr. Collier summons you over to his table. He wants to demonstrate the graph of a fourth-degree polynomial function, but the batteries in his graphing calculator have run out of juice. Explain to Dr. Collier how to create a rough sketch of a graph of a fourth-degree polynomial function.

4.Mrs. Collins is at the table with you and states that the fourth-degree graphs she has seen have four real zeros. She asks you if it is possible to create a fourth-degree polynomial with only two real zeros. Demonstrate how to do this and explain your steps.

Answer 1

1.) By the factor theorem, x - a is a factor of f(x) if f(a) = 0.
For the given function, f(-2) = f(3) = f(-4) = 0. Therefore, x + 2, x - 3 and x + 4 are factors of f(x).

2.) If division by x + 2 gives a remainder of 0, then -2 is a zero of g(x) not 2.

3.) The graph of a fourth degree polynomial starts at a point and turns three times before ending on the same sides that it started.

4.) A fourth degree polynomial can have two real zeros when one of the zeros has 3 multiplicities and the other zero, one multiplicities or when both zeros has 2 multiplicities.