Question

Write a polynomial of the smallest degree with roots 1, -3, and 4.
PLEASE HELP. i understand the basics of polynomials, but not how to do this:/ i will give a medal and fan whoever can show me how to do this AND give me the correct answer. Thank you in advance to whoever responds:) ALSO: IF YOU ARE WILLING TO HELP ME IN THE NEAR FUTURE WITH MORE PROBLEMS SEND ME A MESSAGE. please and thank you again.

Answer 1

3 roots implies 3rd degree. write the roots as linear equations...

x=1 => (x-1)

then multiply those together.

Answer 2

The smallest degree would be 3, because that is the simplest way to guarantee 3 factors. If you know the roots, you can put it as (x - 1)(x - -3)(x - 4). Each of these will give you a root (so y will equal zero). Simplifying, you get
(x-1)(x+3)(x-4). I think this should be a good form, but you could put it as x^3-2x^2-11x+12

Answer 3

So, the next one says to do the same with the roots -8. 9. 10. -11; so in that case, the smallest degree would be four? also thank you very much:)

Answer 4

Yes. So, for each root, you have a factor (x-#)

Answer 5

okay, does it matter which order i put the numbers in? and are the roots they give me the ^#s? like x^8 for the -8?

Answer 6

and ahead of time, i apologize for all the questions. lol

Answer 7

no. So, for the x=-8 root, the factor will be (x+8).
For the x=9 root, the factor will be (x-9)
So, the degree will be the number of roots. You have (x+8)(x-9)(x-10)(x+11), so when you wet it equal to zero, you can find the roots. For example, you have
(x+8)(x-9)(x-10)(x-11)=0
If one of the factors is zero, it all will equal zero. Like, if we have 1 x 5 x 6 x 42 x 10002354 x 0, all that matters is that one zero, since it makes it all 0. So, if we set each factor equal to 0 and solve, we get:
x+8 = 0 --> x=-8
x-9 = 0 --> x=9
x-10 = 0 --> x=10
x+11 = 0 --> x=-11
Does this make more sense?

Answer 8

OH! so in the end it all has to equal zero every time? so the polynomial would be the thing you put above with all the parenthesis? so, if the roots were....oh, idk, 7, -3, 8, and 12 then i would say:
x+7=0 so x=-7
x-3+0 so x= 3

and so on.
which would make my polynomial (x-7)(x+3)(x-8)(x-12) ? and the smallest degree would be 4?

Answer 9

That is 100% CORRECT! I'm glad you understood.

Answer 10

also, should i always end my polynomials with =0? And THANK YOU SO MUCH! i might have a chance at this now! lol

Answer 11

oh one more question if you have the time?

Answer 12

You don't have to always end with =0, but that is how you find the roots. I have time.

Answer 13

oh, okay lol and thanks. so my next problem is:
Write a polynomial of the smallest degree with roots 4i and -4i.
so the smallest degree would be 2 (right?) and then i would write it like (x-4i)(x+4i) ?
the letters have me a little bit confused. :o

Answer 14

Yes, that is right. Do you know about imaginary numbers?

Answer 15

and then my last one on this assignment is
Write a polynomial of the smallest degree with roots 6, 3i, and -3i.
so it would be (x-6)(x-3i)(x+3i) ? and no not really,, im honestly pretty sh*tty with pre calc and trig. lol im lucky to be passing right now.

Answer 16

You're doing well. You are absolutely right. Usually, though, the (x-3i)(x+3i) terms will be multiplied to be (x^2+9), to eliminate the imaginary numbers.

Answer 17

brb gotta go eat. thank you for all of your help so far, and if youre still on when i get back i have a couple more questions regarding some of the info you put in the beginning of this all.

Answer 18

I probably won't be. Sorry!