Question

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.
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). Partygoers offer up solutions, and it is your job to find the right ones.
Their suggestions are:

Answer 1

• (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 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.
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 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

@ParthKohli

Answer 3

@hba

Answer 4

Have you tried plugging in the possible solutions?

Answer 5

i already got x+4 x+2 x-3 @ParthKohli

Answer 6

What's the problem then?

Answer 7

The rest of it, lol

Answer 8

lol

Answer 9

Here's a subtle hint.
If you have a polynomial f(x)
and dividing it by a polynomial (x - r) gives a zero remainder, then that means r is a zero of the polynomial f(x).

What does that say about Professor McCoy's claim?

Answer 10

I'm still stuck on # 1 LOL @Supreme_Kurt

Answer 11

Thought you already got it?

Answer 12

yeah but idk how to justify the selections with complete sentences.

Answer 13

The usual problems :)
There are several ways to determine whether a binomial x - r is indeed a factor of a polynomial.

For number 1, we may as well stick to the simplest.

Each binomial x-r has a root, and it is r.

For instance, in the binomial x - 4, its root is 4.
For x - 2, its root is 2.

Careful, though, for x + 2, the root is -2. Mind your signs.

Everything understood?

Answer 14

nvm..

Answer 15

Something wrong?

Answer 16

not my question.

Answer 17

We'll get to that. I'm trying to explain the concept behind it :)

Answer 18

oh okay sorry yes i got that so far :)

Answer 19

but i'm confused on why x+2 root would be -2

Answer 20

Good. So, we'll test out your choices to see if they are indeed, factors of your polynomial.

What's the root of x+4 ?

Answer 21

Because in any x - r, the root is r itself.

And x + 2
is the same as x - (-2)
and so, the root is -2

Answer 22

oh so the root of x+4 is 4

Answer 23

You can also (and should) think of the root as the value of x that would make the polynomial equal to zero.
Notice that in x + 2, the value of x that would make x+2 zero is -2.

You may want to rethink that.

Answer 24

oh yeaahhhh -4 because x+4 is same as x-(-4)

Answer 25

Yes. Exactly.
So, x+4 is a factor of your polynomial
x3 + 3x2 – 10x – 24

IF -4 is ALSO a root of this longer polynomial.
So... test it out.

See if substituting -4 for x in
x3 + 3x2 – 10x – 24
results in zero.

Try it. I'll wait ^^

Answer 26

-4^3+3(-4)^2-10(-4)-24?

Answer 27

...yes.
Go ahead and do the dirty arithmetic work. You can do it :D

Answer 28

Stuck?

Answer 29

thank you!
-64-144+40-24

Answer 30

i think i got it @Supreme_Kurt

Answer 31

144 is off. Try figuring out 3(-4)^2 again :)

Answer 32

i got +144

Answer 33

Still no. Remember, you only square the (-4) part, not the 3. ;)

Answer 34

oooh yeaaah 48 sorry

Answer 35

@Supreme_Kurt

Answer 36

Good. Now simplify ^^

Answer 37

I got 0

Answer 38

So that proves that x+4 is, in fact, a factor.

Try it with x+3.
What's the root of x+3 ?

Answer 39

3^3+3(3)^2-10x-24

Answer 40

sorry -10(3)

Answer 41

27+27-30-24

Answer 42

Is it zero?

Answer 43

and I also got 0 for that yeah

Answer 44

So, that means x+3 is a factor :D

Answer 45

You getting the hang of it yet? :D

Answer 46

yes i am :)

Answer 47

Oops, I meant x - 3
My bad haha

Answer 48

What's the root of x+2?

Answer 49

-2

Answer 50

Good. Now try -2 with your long polynomial, see it gives zero :)

Answer 51

8+12+20-24

Answer 52

no i got 16

Answer 53

Careful... what's (-2)^3 ?

Answer 54

-8 sorry

Answer 55

Good. Try again ;)
Be VERY careful with signs. Just one wrong sign can really f**k things up... trust me :)

Answer 56

lol yeh so now i got 0

Answer 57

And that wraps it up :D

Answer 58

so is that how i should explain in complete sentences?

Answer 59

Yes. For instance, x+4 is a factor of the polynomial f(x) because f(-4) is zero.
Then show your work.

Same for the other factors :D

Answer 60

okay thanks