SOLUTION: Find the cubic wquation that has -1 adn 2i as roots. Choices a)y=x cubed - x squared + 4x - 4 b)y=x cubed + x squared - 4x - 4 c)y=x cubed + x squared + 4x + 1 d)y=x cubed + x

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find the cubic wquation that has -1 adn 2i as roots. Choices a)y=x cubed - x squared + 4x - 4 b)y=x cubed + x squared - 4x - 4 c)y=x cubed + x squared + 4x + 1 d)y=x cubed + x       Log On


   

Question 963221: Find the cubic wquation that has -1 adn 2i as roots. Choices
a)y=x cubed - x squared + 4x - 4
b)y=x cubed + x squared - 4x - 4
c)y=x cubed + x squared + 4x + 1
d)y=x cubed + x squared + 4x + 4
e)y=x cubed - x squared + 4x - 1
The book says the answer is D, but I do not understand the steps to get there.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

the cubic equation that has x%5B1%5D=-1 and x%5B2%5D=2i as roots, have also x%5B3%5D=-2i because complex root come always in pairs
now, since we have all roots, we can use zero product rule to see which cubic equation is right answer
y=%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29
y=%28x-%28-1%29%29%28x-2i%29%28x-%28-2i%29%29
y=%28x%2B1%29%28x-2i%29%28x%2B2i%29
y=%28x%2B1%29%28x%5E2-%282i%29%5E2%29
y=%28x%2B1%29%28x%5E2-%284i%5E2%29%29
y=%28x%2B1%29%28x%5E2-%284%28-1%29%29%29
y=%28x%2B1%29%28x%5E2-%28-4%29%29
y=%28x%2B1%29%28x%5E2%2B4%29
y=x%5E3%2B4x%2Bx%5E2%2B4
y=x%5E3%2Bx%5E2%2B4x%2B4....=> so, answer is D
or, you can do it this way:
y=x%5E3%2B+x%5E2+%2B+4x+%2B+4....to find zeros, set y=0 and factor completely
0=%28x%5E3%2B+4x%29%2B+%28x%5E2++%2B+4%29
0=x%28x%5E2%2B+4%29%2B+%28x%5E2++%2B+4%29
%28x%2B+1%29+%28x%5E2++%2B+4%29=0
solutions:
if %28x%2B+1%29 => x=-1
if %28x%5E2++%2B+4%29=0 => x%5E2+=-+4=>x=sqrt%28-4%29=>x=2i or x=-2i
=> one more time we got that answer is D