Question

I need help with algebra II

Answer 1

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.

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). Partygoers 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.

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 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 partygoers in full and complete sentences. Make sure you reference any theorems that support your justifications.

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.
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.

Answer 2

Insteresting reading!
Which part do you have a problem with?

Answer 3

all of it

Answer 4

I would like you to start with the first part, and explain to me what you think, or how you would approach it.

Answer 5

i think i can answer 3 and 4 but i and 2 is hard

Answer 6

ok, that's good, shall we start with the first part?

Answer 7

yes

Answer 8

... I'm listening...

Answer 9

i know I have to use the synthatic division for the first one to find my answers

Answer 10

Correct, but only if you find the first factor.
How do you propose to find the first factor?

Answer 11

by listing some numbers pick one and plug it into the formula if it equals 0 thenn thats your answer

Answer 12

Exactly! This is applying the factor theorem, which says that
if (x-k) is a factor of f(x), then f(k)=0.

now have you seen Descartes rule of signs?

Answer 13

yes

Answer 14

How many positive roots are there in f(x)=0?

Answer 15

2

Answer 16

Well, the coefficients of f(x) changed signs once, so there is exactly 1 real positive root, since 1 single root cannot be complex, agree?

Answer 17

oh ok

Answer 18

So far so good?
Now can you go ahead and check/test the given factors to see which one(s) is/are factors of f(x), by evaluating f(x). If f(x)=0, then you found a factor.
Remember, if (x-k) is a factor, then f(k)=0 (watch the sign).
Ready...go!

Answer 19

i got x=3 as the root so x-3 is the factor. so i plugged 3 into the equation and I got this

Answer 20

f(3)=(3)^3+3(3)^2-10(3)-24

Answer 21

Great, yes, indeed, f(3)=0 so (x-3) is a factor.
Can you now proceed to do synthetic division to reduce the cubic to a quadratic?

Answer 22

I got x^2+6x+8

Answer 23

Sure you can factorize this quadratic, right?

Answer 24

then i split the middle tern and got x^2+2x+4x+8 and got x(x+2)+4(x+2) to equal (x+2)(x+4). so my factors for this question are (x-3)(x+2) and (x+4)

Answer 25

f(x)=(x-3)(x+2)(x+4)

Answer 26

Great! You just got the complete solution for part 1! congrats!

Answer 27

really?

Answer 28

Yep!
In fact, I think you're ready for part 2, it takes much the same background as part 1.

Answer 29

could you walk me through step 2

Answer 30

Recall that if (x-k) is a factor, then f(k)=0, right?

Answer 31

yes

Answer 32

Now you are the judge, and give a judgment for each of
McCoy, Guerra, and Romano.
How's that?

Answer 33

good i think

Answer 34

I think the last 2 are correct but the first one is wrong because 2 will be zero if(x-2) is a factor and not (x+2).

Answer 35

McCoy is definitely wrong as you correctly reasoned.
Can you explain to me about the other two?

Answer 36

i think ms. guerra and romano are correct with their explanations

Answer 37

ok, that's good!

Answer 38

Can you show me what you think of #3?

Answer 39

I dont think number 3 has a specifc answer because there is no function to solve, but i could say we will find zero's of given fourth-degree polynomial function and mark those points on x-axis and make a rough sketch of the function passing through those points.

Answer 40

There is a little trick for plotting polynomials.
If the polynomial has a positive leading coefficient, then even-degree polynomials look like a letter W (i.e. opens up, starts from +inf, and ends in +inf). An odd-degree polynomial would start at -inf and end up at +inf.
If the leading coefficient is negative, then everything will reflect about the x-axis.

Answer 41

oh

Answer 42

Also for plotting purposes, you may want to equate the derivative to zero to find the number and location of local extrema, which helps you to do the plotting.
In addition to the x-intercepts that you mentioned, the y-intercept is also quite helpful in plotting.

Answer 43

I think this more or less wraps up the whole problem.

Answer 44

Do you have other points I could help you clear up?