anonymous
  • anonymous
how do you find the zeros of f(x)=x^4+5x^3+10x^2+20x+24?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
wolfram alpha
anonymous
  • anonymous
or try some negative numbers that divide \(24\) the zeros clearly cannot be positive right?
Psymon
  • Psymon
For this we have to use something called the rational zeros theorem. What that requires is that we take the factors of the constant number, 24 in this case, and divide it by all the factors of the leading coefficient (the number in front of the highest power of x), which is 1 in this case. When we do this, our results are all of our POSSIBLE zeros of the polynomial. So we set it up like this: \[\frac{ \pm 24 }{ \pm 1}\] doing this means all of our possible zeros are \[\pm1, \pm2, \pm3, \pm4, \pm6, \pm12, \pm24 \]Now this is a lot of possiblities, but when we pick one it can help us get an idea if we need to guess higher or lower. This make sense so far?

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anonymous
  • anonymous
Descartes rules? am I supposed to find factors of 24 and 1?
Psymon
  • Psymon
Correct :3 So basically what I have done above.
anonymous
  • anonymous
so far so good
Psymon
  • Psymon
Okay, awesome. Now I have a general idea what the answer will be, so I am going to choose -3 as our number to check. Once we choose a number, we do synthetic division and see if the answer turns out to have no remainder. You know how to do synthetic division?
anonymous
  • anonymous
yes i do, i got 390 so this cant be a root
Psymon
  • Psymon
Ah. You must have done something wrong, I do show it to be a root. Do you want to see my work?
anonymous
  • anonymous
yes please
anonymous
  • anonymous
oh i got it i put in a +3 by mistake
Psymon
  • Psymon
|dw:1378173968267:dw|
Psymon
  • Psymon
Lol, no problem xD
Psymon
  • Psymon
So we have shown that -3 is a zero. Or you could say that (x+3) is a factor, either or. Now we would need to once again do our rational zero's check. So as you showed that you know, we would do factors of 8 divided by factors of 1. This time for our synthetic division, I would choose -2.
anonymous
  • anonymous
I thought when using synthetic division you have to get the the inverse of the divisor
Psymon
  • Psymon
What do you mean?
anonymous
  • anonymous
oh nvm Im thinking of something else, sorry bout that
Psymon
  • Psymon
Well, if we were to do it the long division way, I would have done: |dw:1378174220971:dw| But remember, when we have things in factored form, we have to know that the zero is opposite of the actual sign given. So if a factor is (x+3), then the zero is -3. If a factor is (x-6), then the zero is positive 6. When doing synthetic division, we use the value of the actual zero and not the value of what would be paired with x in factored form.
anonymous
  • anonymous
ooh thats what i was getting confused with. So if I find a zero am I suppose to just check everything or can I cross of a section from the list?
Psymon
  • Psymon
Well, we just found that -3 is a zero. So we can either say x = -3, ___, ___, ___ and then find our remaining zeros or we can keep it in factored form and say that (x+3) is a factor. Just depends on what the problem is asking you. Sincer it seems you need zeros, I would just say so far that weknow x = -3.
Psymon
  • Psymon
Then we just keep doing the same process with our synthetic division and factoring until we do not need to anymore.
anonymous
  • anonymous
so got -3 and -2 as the only zeros
Psymon
  • Psymon
Well, there are two complex zeros. What you should have remaining is x^2 + 4, correct?
anonymous
  • anonymous
hmm not sure where your getting that from
Psymon
  • Psymon
Ill check, I may have remembered wrong what I found earlier xD
Psymon
  • Psymon
Yes, after you do synthetic division a second time with -2, you have a remainder of x^2+4
anonymous
  • anonymous
|dw:1378175076210:dw| I am confused b/c this is what I got in my 3rd row