Question

can somebody please help me i need help now please. ill fa you and give you a medal.

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

@dumbcow can you help

Answer 3

i will ok

Answer 4

@Phebe thaks

Answer 5

lets start with #1

Answer 6

wlc

Answer 7

ok

Answer 8

what do we do @Phebe

Answer 9

@mathstudent55

Answer 10

@Phebe do u kno

Answer 11

no sorry but wats #1 Q

Answer 12

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.

Answer 13

@mathstudent55 please help

Answer 14

ok orry im busy but if u need anything ele tag or pm me k

Answer 15

k

Answer 16

Hello. Are you familiar with "grouping"?

Answer 17

x^3 + 3x^2 - 10x + 24 is your polynomial. And one thing you might try to begin with is to "group" the pairs of terms like this: (x^3 - 3x^2) + (-10x + 24)

Answer 18

Then what you want to do is look at those expressions inside the parentheses. So, looking first at (x^3 - 3x^2), is there a factor that x^3 and 3x^2 have in common?

Answer 19

Something that you could divide out of both of those terms?

Answer 20

oh im back @anteater let me catch up here

Answer 21

Ok :)

Answer 22

@anteater i stink at math can u help me so what do we do for #1

Answer 23

Well, we can start by grouping the terms as I did: (x^3 - 3x^2) + (-10x + 24). Then we look at the terms inside those parentheses and see if they have a common factor that can be divided out. So, for the first pair, x^3 and -3x^2, you could divide an x^2 out of both of those. So we can rewrite your polynomial like this:
x^2(x - 3) + (-10x + 24)

Answer 24

Then we also want to look at the terms inside the second set of parentheses, -10x and 24. Is there a number or expression you could divide out of both of those?

Answer 25

2

Answer 26

Yep :) So we can try that. If you were to divide out a 2, we would then have:

x^2(x - 3) + 2(-5x + 12). Now, what I was hoping to see was that after we grouped and factored out common factors that the expressions in side the parentheses would be the same. Unfortunately, they are not, so regrouping that way didn't work. But, there are other things I can try. Would you mind if I take a moment to see if I can come up with the factorization on paper?

Answer 27

sure go ahead thanks

Answer 28

take ur time

Answer 29

Thanks! I will try to be quick.

Answer 30

dont worry

Answer 31

Thank you for waiting!

Answer 32

NO PROBLEM

Answer 33

Please tell me if you have studied something called "rational roots theorem"

Answer 34

yes but i dont get it

Answer 35

It is a method for coming up with a list of possible roots, which will allow you to find the factors of your polynomial. It is used on polynomials of degree 3 and higher when you can't see a straightforward way to factor.

Answer 36

oh okay

Answer 37

What you want to do is look at the coefficients (number parts) of the lead term (the x^3 term) and the final term (-24).

Answer 38

And list them. The coefficient of x^3 is just 1, since 1x^3 = x^3

Answer 39

And the only factor of 1 is 1 itself.

Answer 40

okay

Answer 41

So that does simplify this problem somewhat. :)
Now, we also want to list factors of -24.

Answer 42

okay 12 and 2

Answer 43

6 4

Answer 44

Yes, and a bunch more :)

Answer 45

8 and 3

Answer 46

So if we were to write them all out in order, we would have 1,2,3,4,6,8,12 and 24

Answer 47

both positive and negative.

Answer 48

yes

Answer 49

And so then we use those two sets of numbers (the factors from -24 and the factors from 1) to create the list of possible solutions. The solutions to try are ratios between the numbers in the second set and the numbers in the first set. So, 1/1 (or 1), -1/1 (or -1), 2/1 (or 2), -2/1 (or -2), 3/1 (or 3), -3/1 (or -3) , and so forth.

Answer 50

And we would end up with 16 possible solutions, which sounds like a lot.

Answer 51

But, when you consider that there are an infinite number of integers, we have narrowed the selection down considerably! :)

Answer 52

So we could choose one of our 16 numbers to try, one way we could test it is by using synthetic division. Have you done much work with synthetic division (also called synthetic substitution)?

Answer 53

no

Answer 54

Give me just a moment, and I will give an example of how it works. This will take me a couple of minutes, but I will try to be quick.