SOLUTION: Write a polynomial with rational coefficients so that P(x)=0 has the given roots: -1 and -2i?

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Question 1005610: Write a polynomial with rational coefficients so that P(x)=0 has the given roots: -1 and -2i?
Found 2 solutions by Boreal, MathLover1:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
roots are -1 and +/- 2i, because complex roots are conjugate. This is a cubic equation.
the factors are (x+1) and (x^2+4), since x^2+4=0, and the roots are +/- 2i
the polynomial is x^3+x^2+4x+4=f(x)
graph%28300%2C200%2C-10%2C10%2C-10%2C10%2Cx%5E3%2Bx%5E2%2B4x%2B4%29

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

P%28x%29=0 has the given roots:
x%5B1%5D=-1 and x%5B2%5D=-2i
if we have x%5B2%5D=-2i then we also have x%5B3%5D=2i because complex roots always come in pairs
use zero product formula
P%28x%29=%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29
P%28x%29=%28x-%28-1%29%29%28x-%28-2i%29%29%28x-2i%29
P%28x%29=%28x%2B1%29%28x%2B2i%29%28x-2i%29
P%28x%29=%28x%2B1%29%28x%5E2-%282i%29%5E2%29
P%28x%29=%28x%2B1%29%28x%5E2-4i%5E2%29
P%28x%29=%28x%2B1%29%28x%5E2-4%28-1%29%29
P%28x%29=%28x%2B1%29%28x%5E2%2B4%29
P%28x%29=x%5E3%2B4x%2Bx%5E2%2B4
highlight%28P%28x%29=x%5E3%2Bx%5E2%2B4x%2B4%29
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3%2Bx%5E2%2B4x%2B4%29+