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anonymous

  • one year ago

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

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  1. anonymous
    • one year ago
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    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.

  2. anonymous
    • one year ago
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    \(\frac{p}{q}\) must be written in the lowest term.

  3. anonymous
    • one year ago
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    well, you have a cubic polynomial with integer coefficients:\[x^3 - 15x^2 + 73x -111=0\]

  4. anonymous
    • one year ago
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    constant term: \(-111\) leading coefficient: \(1\)

  5. anonymous
    • one year ago
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    does it have to do something with multiples of those numbers?

  6. anonymous
    • one year ago
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    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.

  7. anonymous
    • one year ago
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    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 \(-1, 1, 3, -3\)

  8. anonymous
    • one year ago
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    If you test those numbers, you can see that \(x=3\) is a solution

  9. anonymous
    • one year ago
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    Now try to factor out \((x-3)\) and the rest will be a quadratic, which will give you other two solutions.

  10. anonymous
    • one year ago
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    How do you factor it out of it? I'm not sure if I'm doing it right. @mukushla

  11. anonymous
    • one year ago
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    hint:\[x^3 - 15x^2 + 73x -111=x^3-3x^2-12x^2+36x+37x-111 \]

  12. anonymous
    • one year ago
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    so it's x^2 -12x +37 and x-3 and after i plug the first equation into the quadratic formula?

  13. anonymous
    • one year ago
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    right

  14. anonymous
    • one year ago
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    i got 6 + √ -4 and 6 - √ 4 not 6 + i and 6-i

  15. anonymous
    • one year ago
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    i meant 6 - √ -4 and 6 + √ -4

  16. anonymous
    • one year ago
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    wait actually i forgot to simplify √ -4 first before i divided by 2

  17. anonymous
    • one year ago
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    i think i know what i did wrong

  18. anonymous
    • one year ago
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    aha, ok

  19. anonymous
    • one year ago
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    Thanks for all your help! :)

  20. anonymous
    • one year ago
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    no problem

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