anonymous
  • anonymous
Find all of the zeros of the function x^3 - 15x^2 + 73x -111 How would you do this? The answer is 3, 6-i, 6+i. I can't remember how to solve it though. @mukushla
Mathematics
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Hi, If you remember rational root theorem, it says possible rational roots of a polynomial with integer coefficients are in the form \(\frac{p}{q}\) where \(p\) is a divisor of constant term and \(q\) is a divisor of leading coefficient.
anonymous
  • anonymous
\(\frac{p}{q}\) must be written in the lowest term.
anonymous
  • anonymous
well, you have a cubic polynomial with integer coefficients:\[x^3 - 15x^2 + 73x -111=0\]

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anonymous
  • anonymous
constant term: \(-111\) leading coefficient: \(1\)
anonymous
  • anonymous
does it have to do something with multiples of those numbers?
anonymous
  • anonymous
o yeah, actually when you want to find the roots of a cubic polynomial by hand, you should look for a integer solution (a small integer number usually) for it in order for factoring the polynomial.
anonymous
  • anonymous
rational root theorem helps you to find that root, now if you apply the theorem some of small possible roots that can be made in the form \(\frac{p}{q}\) are\[\frac{\pm1}{\pm1}, \frac{\pm3}{\pm1}, \frac{\pm37}{\pm1}, ...\]some of the smallest ones are \(-1, 1, 3, -3\)
anonymous
  • anonymous
If you test those numbers, you can see that \(x=3\) is a solution
anonymous
  • anonymous
Now try to factor out \((x-3)\) and the rest will be a quadratic, which will give you other two solutions.
anonymous
  • anonymous
How do you factor it out of it? I'm not sure if I'm doing it right. @mukushla
anonymous
  • anonymous
hint:\[x^3 - 15x^2 + 73x -111=x^3-3x^2-12x^2+36x+37x-111 \]
anonymous
  • anonymous
so it's x^2 -12x +37 and x-3 and after i plug the first equation into the quadratic formula?
anonymous
  • anonymous
right
anonymous
  • anonymous
i got 6 + √ -4 and 6 - √ 4 not 6 + i and 6-i
anonymous
  • anonymous
i meant 6 - √ -4 and 6 + √ -4
anonymous
  • anonymous
wait actually i forgot to simplify √ -4 first before i divided by 2
anonymous
  • anonymous
i think i know what i did wrong
anonymous
  • anonymous
aha, ok
anonymous
  • anonymous
Thanks for all your help! :)
anonymous
  • anonymous
no problem

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