SOLUTION: Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing​ utility, use it to graph the function and v

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing​ utility, use it to graph the function and v      Log On


   

Question 1115284: Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing​ utility, use it to graph the function and verify the real zeros and the given function value.
n=3,
-3 and 6+5i are zeros;
f(1)=200
Please help I am completely lost here and have no idea how to solve this :( :(
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
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n value.
n=3,
-3 and 6+5i are zeros;
f(1)=200
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Three factors with x needed, because degree 3, or n=3.

y=a%28x-h%29%28x-k%29%28x-m%29

You are given two of the zeros, and since complex zeros come as "conjugate pairs", you can fill in form your three given zeros.

y=a%28x%2B3%29%28x-%286%2B5i%29%29%28x-%286-5i%29%29
Simplify this, at least as far as maybe to get real number coefficients.
y=a%28x%2B3%29%28%28x-6%29-5i%29%28%28x-6%29%2B5i%29
y=a%28x%2B3%29%28%28x-6%29%5E2-%2825i%5E2%29%29
y=a%28x%2B3%29%28x%5E2-12x%2B36%2B25%29
highlight_green%28y=a%28x%2B3%29%28x%5E2-12x%2B61%29%29------still in factored form, but showing only real coefficients in both factors.

Find the unknown factor, a.
f%281%29=200=a%281%2B3%29%281%5E2-12%2A1%2B61%29
4%2A%281-12%2B61%29a=200
4%2862-12%29a=200
4%2A50a=200
a=200%2F200=1-----this factor is 1.

highlight%28y=%28x%2B3%29%28x%5E2-12x%2B61%29%29--------------the equation you want, unless you wish it multiplied further.

Use whatever graphing tool you want to.