Question 199809: Hi!
Find a polynomial with integer coefficients and a leading coefficient of one that satisfies the given conditions.
P has degree 3, and zeros 3 and 1 + 4i.
thanks for your help!
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
If the polynomial is of degree 3, then there are guaranteed to be three zeros, at least one of which is a real number zero. You are given one real number zero and one complex number zero. But the thing we also know is that complex zeros always come in pairs, and those pairs are always conjugates. That means that if  is a zero, then  , the conjugate, is also a zero.
You are given:
 and
 , therefore
 must also be a zero.
Knowing the zeros of a polynomial allows us to determine the 1st degree factors of the polynomial:
 , so
 is a factor.
Likewise:
) and )
are factors, so the polynomial can be determined from the product of the three factors:
(x - (1 + 4i))(x - (1 - 4i)))
When you multiply these, do the complex factors first. Hints:
1. Treat  and  as single numbers.
2. times results in the difference of two squares
3. Remember 
John
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