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):
Answer by Edwin McCravy(20056)  (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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