anonymous
  • anonymous
Find a polynomial function of least degree having only real coefficients with zeros as given. -3,2,-i, and 2+i
Mathematics
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schrodinger
  • schrodinger
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anonymous
  • anonymous
mrsheinold, if these are the zeros of your polynomial P(z) for z complex, then you have\[P(z)=(x+3)(z-(2-i))(z-(2+i))\]All you need to do is expand this out to get your polynomial in a more amenable form:\[P(z)=z^3-z^2-7z+15\]
anonymous
  • anonymous
You'll always have real coefficients if your zeros are real or come in complex conjugate pairs, like you have...
anonymous
  • anonymous
Ok... I dont understand. The example in the book uses f(x) not P(z) im assuming thats ok.. but when I got my formulas at first I got f(x)=(x+3)(x-2)(x+i)(x-2-i) how did you get your first formula??

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anonymous
  • anonymous
f(x), P(z)...semantics, don't worry about it. Use what the book uses. You get the first formula by the following: if \[\alpha, \beta, \gamma\]are roots of a polynomial, f(x), then \[f(x)=(x-{\alpha})(x-\beta)(x-\gamma)\]All you need to do is sub. your roots in.
anonymous
  • anonymous
They are defined as roots of the equation because when x takes any of those values, one of the factors becomes zero, giving f(x)=0.
anonymous
  • anonymous
Is this clearer?
anonymous
  • anonymous
Not really. I have 4 numbers -3,2,-i, 2+i I'm suppose to change the signs and put them in () So I thought -3 would change to (x+3) and the 2 would change to (x-2) -i would change to (x+i) and 2+i would change to (x-2-i) are those right or wrong??
anonymous
  • anonymous
Oh sorry bud, I read it as 2-i, not 2,-i...like I said, I'm ready for bed. Hang on...
anonymous
  • anonymous
mrsheinold, since you are given 4 numbers two of which are complex, your desired polynomial should have 6 factors, P(x)=(x+3)(x-2)(x-i)(x+i)(x-(2+i))(x-(2-i)) as stated above when you are given roots of a polynomial that are complex, those roots will appear in conjugate pairs. e.g. find the roots of P(x)=x^2+1

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