SOLUTION: hi! A polynomial P is given. P(x) = x^3 + 216 (a) Find all zeros of P, real and complex. (b) Factor P completely. thanks for the homework help!

Question 199846: hi!
A polynomial P is given.
P(x) = x^3 + 216
(a) Find all zeros of P, real and complex.
(b) Factor P completely.
thanks for the homework help!
Answer by RAY100(1637) (Show Source): You can put this solution on YOUR website!
great problem
.
to factor,,,,( x^3 +216),,,,first examine the last term for factors, 1,216, 2,108,3, 72, 4, 54, 6,36
,
now we try to find which factor will go evenly into the function.,,,,usually we do this by division, either polynomial or synthetic.
.
using synthetic division on a best quess of (x+6) factor or x=-6 zero:
.
-6,,,,,,,,,,,,,,1 0 0 216,,,,,,,,,,,,,,,,,,remember to fill for x^2 and x terms
,,,,,,,,,,,,,,,,,,,-6 36 -216
______________________________
,,,,,,,,,,,,,,1,,,-6,,,36,,,,,,,,0,,,,,,,,,,,,,,,as remainder is zero, this is a factor
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(x+6) ( x^2 -6x +36),,,,,,,,,,,to factor the second term, use the quadratic formula
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a= (1),,,,b=(-6) ,,,,,,c=(36)
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x =[ -(-6) +/- { (-6)^2 - 4 (1) (36) } ^(1/2) ] / 2 ( 1)
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x= [ 6 +/- { 36 -144)^(1/2) } ] /2
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x= [ 6 +/- {-108)^(1/2) } ] /2,,,with sqrt (-108) = sqrt (-1) * sqrt (108 = 10.4 i
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x= 3 +/- ( 5.2 i ), x= 3+5.2i ,,,,,or x = 3-5.2i
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Collecting all the zeros,,,,x= (-6),,,, ( 3+5.2i) ,,,,,(3-5.2i)
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checking
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( x+6) ( x-(3+5.2i) ) ( x-(3-5.2i) )
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(x+6) ( x-3-5.2i) ( x-3+5.2i),,,,,good summary of all factors
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(x+6) ( x^2 -6x +36)
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x^3 + 216,,,,,ok
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