SOLUTION: How do you write a polynomial function of least degree with integral coefficients with these given zeros -1,-2+2i

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Question 1003853: How do you write a polynomial function of least degree with integral coefficients with these given zeros
-1,-2+2i
Found 2 solutions by josgarithmetic, MathLover1:
Answer by josgarithmetic(39625) About Me  (Show Source):
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given zeros:
x%5B1%5D=-1,real zero
x%5B2%5D=-2%2B2i, complex zero
recall, complex zeros always coming in pairs; so, this function also have x%5B3%5D=-2-2i as a zero
now, use zero product theorem to find a function that has these three zeros
f%28x%29=%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29
f%28x%29=%28x-%28-1%29%29%28x-%28-2%2B2i%29%29%28x-%28-2-2i%29%29
f%28x%29=%28x%2B1%29%28x%2B2-2i%29%28x%2B2%2B2i%29

f%28x%29=%28x%2B1%29%28x%5E2%2B4x%2B4-4%28-1%29%29
f%28x%29=%28x%2B1%29%28x%5E2%2B4x%2B4%2B4%29
f%28x%29=%28x%2B1%29%28x%5E2%2B4x%2B8%29%29
f%28x%29=x%5E3%2B4x%5E2%2B8x%2Bx%5E2%2B4x%2B8
f%28x%29=x%5E3%2B5x%5E2%2B12x%2B8