SOLUTION: what are all zeros of f(x)=X4th power-X3rd power+3x squared-9X-54

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Question 166990: what are all zeros of f(x)=X4th power-X3rd power+3x squared-9X-54
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let's graph the function first.
+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E4-x%5E3%2B3x%5E2-9x-54%29+
Looks like x=-2 and x=3 are two zeros.
f%28x%29=x%5E4-x%5E3%2B3x%5E2-9x-54
f%28-2%29=%28-2%29%5E4-%28-2%29%5E3%2B3%28-2%29%5E2-9%28-2%29-54
f%28-2%29=16%2B8%2B12%2B18-54
f%28-2%29=0
f%283%29=%283%29%5E4-%283%29%5E3%2B3%283%29%5E2-9%283%29-54
f%283%29=81-27%2B27-27-54
f%283%29=0
Now you have two zeros, you can use synthetic division to find the others.
f%28x%29=x%5E4-x%5E3%2B3x%5E2-9x-54=%28ax%5E2%2Bbx%2Bc%29%28x%2B2%29%28x-3%29
%28x%2B2%29%28x-3%29=x%5E2-x%2B6
ax%5E2%2Bbx%2Bc=%28x%5E4-x%5E3%2B3x%5E2-9x-54%29%2F%28x%5E2-x-6%29
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Showing synthetic division is a little difficult.
The first part will show the factor (left hand column), then the factor times the divisor,
then the next line will show the subtraction.
Repeat.
Hopefully it makes sense.
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..................___________________
(x^2-x-6)|x^4-x^3+3x^2-9x-54
x^2:............x^4-x^3-6x^2
-----------------------------------
.....................................9x^2-9x-54
9:.................................9x^2-9x-54
-----------------------------------
.........................................................0
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%28x%5E4-x%5E3%2B3x%5E2-9x-54%29%2F%28x%5E2-x-6%29=x%5E2%2B9
Now look at the last quadratic and find its roots.
You can factor it.
x%5E2%2B9=%28x-3i%29%28x%2B3i%29
x%5E4-x%5E3%2B3x%5E2-9x-54=%28x%2B2%29%28x-3%29%28x%2B3i%29%28x-3i%29
The zeros are x=-2,3,-3i,3i.