SOLUTION: Given that {{{1+2i}}} and {{{1-2i}}} are zeros of {{{x^4-4x^3+6x^2-4x-15}}}, find the other zeros

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Question 140325: Given that 1%2B2i and 1-2i are zeros of x%5E4-4x%5E3%2B6x%5E2-4x-15, find the other zeros




Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x=1%2B2i or x=1-2i Start with the given zeros


x-1=2i or x-1=-2i Subtract 1 from both sides


%28x-1%29%5E2=4i%5E2 or %28x-1%29%5E2=4i%5E2 Square both sides


%28x-1%29%5E2=4%28-1%29 or %28x-1%29%5E2=4%28-1%29 Replace i%5E2 with -1


%28x-1%29%5E2=-4 or %28x-1%29%5E2=-4 Multiply


%28x-1%29%5E2%2B4=0 or %28x-1%29%5E2%2B4=0 Add 4 to both sides



x%5E2-2x%2B5=0 or x%5E2-2x%2B5=0 Foil and simplify



Now since x%5E2-2x%2B5 is a factor of x%5E4-4x%5E3%2B6x%5E2-4x-15, we can use long division to find another factor





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So another factor of x%5E4-4x%5E3%2B6x%5E2-4x-15 is x%5E2-2x-3




x%5E2-2x-3=0 Set the factor equal to zero

%28x-3%29%28x%2B1%29=0 Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
x-3=0 or x%2B1=0

x=3 or x=-1 Now solve for x in each case




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Answer:



So the other zeros are

x=3 or x=-1