**Module 4 Quiz**

Three partygoers 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 2 is a zero of g(x) because long division with (x+2) results in a remainder of 0.

Ms. Guerra: She says 2 is a zero of g(x) because g(2)=0

Mr. Romano: He says 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 partygoers in full and complete sentences. Make sure you reference any theorems that support your justifications.

My justification

2. The remainder theorem states that when the opposite of the constant from the binomial divisor is substituted into a function for x, the result is the remainder. Basically this means that if given a polynomial function g(x) and a number "a", if (x-a) is a factor of g(x), then "a" is a zero of the polynomial. To prove that the binomial really is a factor of g(x): using long division with (x+2) as the divisor, using synthetic division with -2 as the divisor, or even substituting 2 into g(x) to equal 0 are all plausible scenarios. So, all in all the three partygoers were right in theory. Though Mr. Romano was partially wrong because he would really need to use -2 as the divisor using synthetic division.

3. To graph a fourth-degree polynomial function, you first need to find the x-intercepts. One way of going about this is you could use the Rational Root Theorem. After finding the x-intercepts, the next thing to do is to find the y-intercept. In order to do this you must replace x with 0. Once you have both the intercepts, graph them. Now, because you are trying to graph a fourth-degree polynomial, both ends of the graph will point in the same direction. Whether or not the function is positive or negative will determine which way the ends will point. If it’s positive, the ends will face up and vice versa. From there simply draw connect the dots.

Assessment

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 (see box)

1. The correct factors of f(x) are (x-3)(x+2)(x+4). At first I tried to factor the function by grouping then finding the GCF. However, that didn’t work, but in the process found that (x-3) was definitely a factor. So I synthetically divided x^3+3x^2-10x-24 by x-3, getting x^2+6x+8. Finally I factored the trinomial and retrieved the answer of (x-3)(x+2)(x+4). Giving me my answer.

mysterious box

Mrs. Collins is at the table with you and states that the fourth-degree graphs she has seen have 4 real zeros. She asks you if it is possible to create a fourth-degree polynomial with only 2 real zeros. Demonstrate how to do this and explain your steps.

4. A fourth-degree polynomial can most definitely have only two real zeros. This is possibility due to the fact that the fourth-degree polynomial would also have 2 complex zeros. If a function has complex roots, those complex roots will occur as a conjugate pair. A complex root will have an imaginary "i" in it. If a function has two real zeros, then it will have two complex zeros.

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.

You have been invited to a fancy dinner party celebrating your hard work in Algebra 2 so far. The distinguished guests come from various aspects of math disciplines like professors, engineers, and financial analysts.

f(x) = x3 + 3x2 – 10x – 24. To open the box, you need to identify the correct factors of f(x). Partygoers offer up solutions, and it is your job to find the right ones.