Question 218142
f(x)= x^4 -x^3 +3x^2 -9x -54
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zeros  =p/q= (factors  of constant)/ (factors  of leading  coeff)= +/- 54/1
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or  zeroes  are  factors  of  54,,,+/- 1,54,2,27,3,18,6,9
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If  we  plot  using  a  graphing  calc,  we  find zero's  at  x=-2,,,x=+3
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using  synthetic  division  to  confirm,,  and  find  imaginary  zeros
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.try  x=3
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.,,,,,3|,,,1,,,-1,,,3,,,-9,,,-54
.,,,,,,,,,,,,,,3,,,,6,,,,27,,+54
.,,,,,,,------------------------
.,,,,,,,,,,1,,,2,,,,9,,,18,,,,0,,,ok,,,,now  try x=-2
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,,,,,,-2|,,1,,,2,,,,9,,,18
.,,,,,,,,,,,,,,-2,,,0,,,-18
,,,,,,,,---------------------
,,,,,,,,,,,1,,,0,,,,9,,,,,0,,,,,,,,ok
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Remainder  is  ,,1x^2 +0x +9
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.This  factors  to  (x+3i)(x-3i),,,,,imaginary  zeros,,,(use  quadratic  formula  if you  don't  remember)
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Set  of  zeros  is,,,,x=3,,,x=-2,,,x=+3i,,,,x=-3i
.
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this  can  be  confirmed  by  multiplying  above  to  get  original
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check ,,,,(x^2 +9) (x+2)(x-3) = (x^3 +2x^2 +9x +18)(x-3) = x^4 -x^3 +3x^2 =9x -54,,,,,ok
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