SOLUTION: what is the lowest-degree polynomial with integer coefficients and the roots 6, i, and -i? a. x^3+6x^2-x-6 b. x^3-6x^2-x+6 c. x^3-6x^2+x-6 d. x^3+6x^2+x+6 I think its d?

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Question 636348: what is the lowest-degree polynomial with integer coefficients and the roots 6, i, and -i?
a. x^3+6x^2-x-6
b. x^3-6x^2-x+6
c. x^3-6x^2+x-6
d. x^3+6x^2+x+6
I think its d?
Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


Hint: and are the roots of

John

My calculator said it, I believe it, that settles it
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Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!
what is the lowest-degree polynomial with integer coefficients and the roots 6, i, and -i?
 
It doesn't work to guess.

Set x = to each root:

     x=6,   x=i,   x=-i

Get 0 on the right side of each:

   x-6=0  x-i=0, x+i=0

Indicate the multiplication of all
three left sides and set equal to 0:

    (x-6)(x-i)(x+i) = 0

Multiply two of them (the easiest is the
second and third, since they are conjugates:

       (x-6)(x²-i²) = 0

Use the fact that i²=-1 to replace i²

     (x-6)[x²-(-1)] = 0

        (x-6)(x²+1) = 0

Multiply the left side out:

         x³+x-6x²-6 = 0

Arrange in descending powers of x:

         x³-6x²+x-6 = 0  

So you see that the answer is c., not d.

Edwin
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