Question

polynomial with roots 4, −5, and 7

Answer 1

those are the answers

Answer 2

The "roots" of a polynomial are the x-values of the polynomial (f(x)) such that f(x)=0. There are also called the zeroes of the function (polynomial here). So the quick way to get the answer is to substitute in the x-values into the polynomials and evaluate.

Answer 3

If the result is 0 then that x-value is a root for that polynomial. The answer will be the polynomial that has all three x-values as its roots.

Answer 4

how do you do that exactly?

Answer 5

Depending on your speed with factoring, the longer/shorter method would be to completely factor the polynomials and see which one ends up with the appropriate roots.

Answer 6

Substituting:
\[ f(x)=x^3-6x^2-27x+140\]So:
\[ f(4)=(4)^3-6(4)^2-27(4)+140\]Evaluate.
Rinse and repeat with the other roots and, if necessary, the other polynomials.

Answer 7

The FINAL way and probably the easiest (which is why I left it till last). Is to build the polynomial from its roots.

Answer 8

If a polynomial as all real roots a, b, & c (this one does) then one can write the polynomial as the product: (x-a)(x-b)(x-c).
Now I wrote this only in third order (as per the question), but this is true for any order n.

Answer 9

If you solve the problem this way, then just form the product, multiply it out, then compare to the given choices.

Answer 10

I'm a little confused as to what a,b,c would be though
I know the x's would be replaced with 4,-5,7

Answer 11

the a, b, & c are just standing in for any three numbers that are the roots of the polynomial. Make sense?

Answer 12

By writing (x-a)(x-b)(x-c) I have expressed ALL third order polynomials with REAL roots since a,b, & c are just place holder variables for a given set of 3 real numbers that are the roots of a third order polynomial.

Answer 13

So in this example we could have a=4 b=-5 and c=7

Answer 14

But I want you to appreciate that ANY third order polynomial with real roots can be written this way which is what is so powerful about it.

Answer 15

ohh okay so I was doing this backwards lol

Answer 16

Well there is no wrong way to do this problem. Either way works, but this way seems to be the quickest.

Answer 17

That's why I offered all of the possible ways of solving it. When you sitting there during a test, if you space one way of doing it you can fall back on the others.

Answer 18

So what would the polynomial be? Please post it here before multiplying it out.

Answer 19

Please don't hesitate to try while you have me here mistakes are ok I can quickly correct them.

Answer 20

Alright so I'm trying to work this out and I'm still confused as to how to do it correctly? Could you show me an example

Answer 21

Though generally you should never be afraid to try when it comes to math. If you are wrong that can often be a MORE powerful way of learning. Because it allows you to figure out the error in your thinking.

Answer 22

(I just feel like Im never going to use this stuff again)

Answer 23

Well dont work anything out yet.... Write down the polynomial in the form I have given using the roots given in the problem.

Answer 24

emily its ok to get frustrated at this stuff but dont say that you have no idea what you will ultimately end up doing. I felt the same way when I was younger, and now.... well suffice it to say I use this stuf A L O T!!! :D

Answer 25

Frustration with this stuff is natural just take a breath, and keep trying. Like anything in life practice makes perfect and it takes exercise to get your mental muscle strong.

Answer 26

(x-4)(x+5)(x-7)

I'm sure in my future career I will do something w/o math, but for now I need to pass this class

Answer 27

CORRECT :D :D

Answer 28

Yay! So after this step you said I need to multiply something?

Answer 29

You never know emily I actually went back to school because math related degrees are WAYY more valuable at finding and getting a job. And I ended up needing to know a lot... and I mean a lottttt of math. But I found joy in it because like everything in life, once you get good at it it become really enjoyable. Trust me!

Answer 30

Yes now we will multiply it out.

Answer 31

Ok lets look at the first two terms first: (x-4)(x+5)

Answer 32

can you multiply those out?

Answer 33

alrighty I'm almost done don't go anywhere

Answer 34

Im here

Answer 35

-4*5*-7 = 140 so its either a or d

Answer 36

@welshfella Dont worry I got it

Answer 37

i got 0 I'm pretty sure that's wrong

Answer 38

Ok what happened to multiplying out (x-4)(x+5)

Answer 39

I did!

Ok, let me try this again I'm going to do this differently
(x-4)(x+5)
-4*5
and x stays the same
(x-20)

Answer 40

Do you remember the FOIL technique for multiplying binomials (two term sums in parentheses)? First Outer Inner Last?

Answer 41

I remember that from algebra 1 kinda

Answer 42

but not really

Answer 43

Ok well I will tell you a better way imo of doing this type of stuff. Do you remember the distributive rule?

Answer 44

What is 4(3+1)=?

Answer 45

16

Answer 46

Ok how did you find that?

Answer 47

4(3+1)=
4(4)
16

Answer 48

Ok what about this:
4(3+1)=
4*3+4*1=
12+4=
16
Makes sense right?
I distributed the 4 to both terms inside the parentheses

Answer 49

either way the answer was correct yeah

Answer 50

so what do i need to do?

Answer 51

Ok hold your horses... I want to make sure this is super clear because often times people forget lower level maths and get really frustrated in later level maths thinking it is too hard, but in reality they have just forgotten simple tricks that make the harder problems MUCH MUCH easier.

Answer 52

Ok what about (2+3)(3+5)=?

Answer 53

5+8
13

Answer 54

I wasn't adding them I was multiplying them. Try again.

Answer 55

omg whoops i jumped over that
40 sorry

Answer 56

Its ok... Note I could also DISTRIBUTE :D :D
(2+3)3 + (2+3)5 = 5*3 + 5*5 = 15 + 25 = 40
3*2 + 3*3 + 2*5 + 3*5 = 6 + 9 + 10 + 15 = 40
Notice no matter what level I stop I always get the same answer.

Answer 57

OK i see that, but how do i apply redistribution to what I'm doing ):

Answer 58

Ok so lets do the same with (x-4)(x+5)

Answer 59

:D See now? Do it and post your answer.

Answer 60

Having trouble?

Answer 61

Darn I meant:
(x-4)(x+5)
(x-4)x+(x-4)5

Answer 62

Ok!

(x-4)(x+5)
(x-4)4 + (x-4)5 = 5*-4 + x*x = -20 + x

I tried it like this

Answer 63

I'm sorry but that isnt correct. Please don't skip steps. Think of skipping steps as a privilege you must EARN through mastering the process first. Once you have mastered a concept then you can skip steps (maybe).... but NEVER before. It's alright though just try again.

Answer 64

Just do the first term... what is (x-4)x =

Answer 65

i did it like you showed me on your other comment ):
I'm just so confused
I'm never gonna get this quiz done..

Answer 66

Relax... Relax... Stop for a second. Put down your pencil, close your eyes, and take a few deeps breaths.

Answer 67

Cross your fingers and put your hands on your stomach... Breath deeply and feel your stomach expand and contract.

Answer 68

Do this for a few breaths and try and clear your mind... Do this for a min or so and come back.

Answer 69

When you get back say something so I know.

Answer 70

Ok lets get to it

Answer 71

Feel a little better?

Answer 72

So we are just looking at (x-4)x right?

Answer 73

yes, so i was looking at what you had written earlier
(x-a)(x-b)(x-c)

-4* 5*-7 = 140
5* -7 = -35
-4* 5 = -20
f(x) = x^3 - 20x^2 - 35x +140
I'm thinking this is my answer?

I only have like 10 minutes to finish my quiz, but I'm very confident about my answer please tell me what you think

Answer 74

Oh... umm I had no idea you were just trying to work out the coefficients... hold on.

Answer 75

No sorry its x^3-6x^2-27x+140

Answer 76

i was close /:

Answer 77

After you are done with your quiz plz come back to this posting so we can work out where you are going wrong. It is very important to sort out problems like this because this WILL haunt you again and again if you don't figure out what your doing wrong.

Answer 78

can you help me really fast with one more?
its the opposite of this if you catch my drift

Which of the following represents the zeros of f(x) = 4x3 − 13x2 − 37x + 10?

Answer 79

Can you start a new post please this one is too long already.

Answer 80

gotcha