Question

What polynomial has roots of -5, -4, and 1 ?

Answer 1

a) x3 - 8x2 - 11x + 20
b) x3 - x2 + 22x + 40
c) x3 + x2 - 22x - 40
d) x3 + 8x2 + 11x - 20

Answer 2

If you are given roots of a, b, and c, your polynomial is:

y = (x - a)(x - b)(x - c)

And then just multiply that out if you need to put it into standard form.

Answer 3

So, start by substituting for a, b, and c. and write that out here in Openstudy.

Answer 4

In the next step after you do that, I'll show you how to expand that.

Answer 5

okay

Answer 6

@danny562 ? Where'd you go? Don't you want to know how to do this?

Answer 7

yeah im stuck. i dont get it.

Answer 8

You can't substitute the given zeros into the equation I wrote out? All you do there is everywhere you see an a or b or c, you substitute one of the given numbers. I know you can do that much because anyone can do that. So go ahead and write that out.

Answer 9

@danny562 , you are missing a golden opportunity to learn here. There are tutors here at Openstudy that will give you much more than just an answer (which will not help you at all anyway). They will show you how to get the answer so you can do these problems on your own. You can learn that way, when someone will take the time to teach you. Otherwise, students repeat the same "rut". Looking only to others instead of helping themselves.

Answer 10

no i know! i want to learn but im so confused!

Answer 11

That's a start! That's a big person who admits that and still wants to learn. We can take it from the basics if you need that, no problem! I'll start over with what you need here.

You are given "roots". That translates into factors that you can build that when any one of them is "zero", the "y" value will become "zero". That is the relationship between a root and a zero. In the equation, which I will explain, if any of the factors is zero, the whole thing will be zero:

y = (x - a)(x - b)(x - c)

So, if (x-a) or (x-b) or (x-c) is zero, the whole thing will be zero. Are you with me so far? If not, ask any question.

Answer 12

A little more fundamental:

If you have something like:

(0) times (anything)

that will be zero because zero times anything at all is zero. That's one of the field properties of arithmetic. Are you with me on that?

Answer 13

@danny562 , you might want to answer so that I'll know where to start this for you. I don't want to go too basic or too advanced. I've got to know where you stand in your understanding of math. Otherwise it will be difficult to help you.

Answer 14

Plus, interaction and you trying things along the way is the best way to learn. What do you say? It's up to you. Here, you've got a mathematician willing to go at your pace. It's all up to you.

Answer 15

yes i want to learn

Answer 16

ok, that's good. Back to equation:

y = (x - a)(x - b)(x - c)

This is the same as:

y = (x - a) times (x - b) times (x - c)

but this is bulky so we use the first form. Each expression within parentheses is a factor and when any of the factors is zero, then the whole thing is zero. That makes a, b, and c roots. And x minus "a root" is a factor.

We are given 3 roots so we are going to substitute:

y = [x - (-5)] [x - (-4)] (x - 1)

that is:

y = (x + 5)(x + 4)(x - 1)

because when you subtract a negative, it is the same as adding a positive.

Answer 17

Once we get this far, we are about halfway done, but I'm going to move a little more quickly unless you stop me or slow me down.

y = (x + 5)(x + 4)(x - 1) -> y = [(x + 5)(x + 4)] (x - 1)

y = (x^2 + 5x + 4x + 20) (x - 1)

y = (x^2 + 9x + 20) (x - 1)

Answer 18

y = (x^2 + 9x + 20)x - (x^2 + 9x + 20)

that requires a little explanation and is the result of "breaking apart" the "x - 1" factor. It's the distributive law of multiplication and that is a good place for you to go over and over.

Answer 19

i like it when people explain the whole problem then i can actually undertand it. but then im fine with working on the final part to get the answer

Answer 20

np, I'm getting a feel for where you are.

y = (x^3 + 9x^2 + 20x) - (x^2 + 9x + 20)

y = x^3 + 9x^2 + 20x - x^2 - 9x - 20 getting rid of ( )

y = x^3 + 9x^2 - x^2 + 20x - 9x - 20 getting "like terms" together

y = x^3 + 8x^2 + 11x - 20 adding/subtracting coefficients on like powers

of "x"

Answer 21

And that's your answer in the standard form for a polynomial.

Answer 22

Now, there's a lot of steps and rules and laws. If you have any questions whatsoever, just ask.

Answer 23

oh okay let me see! the answer i got was a) x3 - 8x2 - 11x + 20

Answer 24

am i correct or did i do something wrong?

Answer 25

You are a little off with the signs.

Answer 26

Either we can go over what you did, or you can follow along with what I did and those steps.

Answer 27

so thats not the right answer what did i do wrong on the signs

Answer 28

Well, except for the x^3 term, all the other signs were opposite of what they should be. I don't know what you did wrong, because all I have is your final answer, not your actual work, so I don't know where you went wrong. But it's ok to be wrong, as long as you go back and see where you went wrong so you don't do it again or often. You can either show me what you did or just go over what I did.

Answer 29

It could be back where you substituted the roots. That is probably where you went wrong, but I can't see what you did there. Go back and see what equation I had when I substituted the roots:

y = (x + 5)(x + 4)(x - 1)

Answer 30

Remember, when you create the factors, you have to work with:

x minus the root

So, for root of -5 you have a factor of: x minus negative 5

x - (-5) -> x + 5

Did you get that in your work?

Answer 31

ya

Answer 32

So, if you got that, you probably got to:

y = (x + 5)(x + 4)(x - 1)

just fine. If you didn't, then that's where your problem is.

Answer 33

hey bro thanks for helping me i understand it so much more im on my way to go take my final right now cna you please hoook it up with the answer pleasee!!!!

Answer 34

np, lol! But if you are reading along, the answer is in my 14th post at the end. Just go there and you'll see it.

Answer 35

haha what! 14 post from the end...

Answer 36

Just look for the 14th occurrence of my icon. That posts starts out with:

"np, I'm getting a feel . . ."

Answer 37

oh the answer is d

Answer 38

i see how i got my signs mixed up now thank you so much!!

Answer 39

d.) x3 + 8x2 + 11x - 20

Answer 40

uw! Good luck on your test!

Answer 41

Yep! That's it!