SOLUTION: given f(x)= {{{(x+1)(x^3-27)}}} find all zeros (real and imaginary)

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Question 165304: given f(x)= %28x%2B1%29%28x%5E3-27%29 find all zeros (real and imaginary)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=%28x%2B1%29%28x%5E3-27%29 Start with the given function


0=%28x%2B1%29%28x%5E3-27%29 Plug in f%28x%29=0



0=%28x%2B1%29%28x-3%29%28x%5E2%2B3x%2B9%29 Factor x%5E3-27 to get %28x-3%29%28x%5E2%2B3x%2B9%29. Note: use the difference of cubes formula.


x%2B1=0 or x-3=0 or x%5E2%2B3x%2B9=0 Set each individual factor equal to zero


Solve the first two equations to get x=-1 or x=3. These are the first two zeros


Now let's solve x%5E2%2B3x%2B9=0


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=3, and c=9


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%283%29+%2B-+sqrt%28+%283%29%5E2-4%281%29%289%29+%29%29%2F%282%281%29%29 Plug in a=1, b=3, and c=9


x+=+%28-3+%2B-+sqrt%28+9-4%281%29%289%29+%29%29%2F%282%281%29%29 Square 3 to get 9.


x+=+%28-3+%2B-+sqrt%28+9-36+%29%29%2F%282%281%29%29 Multiply 4%281%29%289%29 to get 36


x+=+%28-3+%2B-+sqrt%28+-27+%29%29%2F%282%281%29%29 Subtract 36 from 9 to get -27


x+=+%28-3+%2B-+sqrt%28+-27+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-3+%2B-+3i%2Asqrt%283%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-3%2B3i%2Asqrt%283%29%29%2F%282%29 or x+=+%28-3-3i%2Asqrt%283%29%29%2F%282%29 Break up the expression.


So the zeros of x%5E2%2B3x%2B9 are x+=+%28-3%2B3i%2Asqrt%283%29%29%2F%282%29 or x+=+%28-3-3i%2Asqrt%283%29%29%2F%282%29


which approximate to x=-1.5%2B2.598i or x=-1.5-2.598i



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Answer:


So in total, the 4 zeros are: x=-1, x=3, x+=+%28-3%2B3i%2Asqrt%283%29%29%2F%282%29 or x+=+%28-3-3i%2Asqrt%283%29%29%2F%282%29


Which in decimal form are x=-1, x=3, x=-1.5%2B2.598i or x=-1.5-2.598i