SOLUTION: Given that -1 is a zero of the polynomial g(x)=x^3-3x^2-3x+1, express g(x) as a product of linear factors. g(x)=

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Question 319327: Given that -1 is a zero of the polynomial g(x)=x^3-3x^2-3x+1, express g(x) as a product of linear factors.
g(x)=
Found 3 solutions by stanbon, solver91311, Edwin McCravy:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Given that -1 is a zero of the polynomial g(x)=x^3-3x^2-3x+1, express g(x) as a product of linear factors.
------------
If -1 is a zero, use synthetic division to find the other factors:
----
-1)....1....-3....-3....1
.......1....-4....1....|..0
-----------
The quadratic factor is x^2-4x+1
---
Use the quadratic formula to find the zeroes:
x = [4 +- sqrt(16-4*1)]/2
x = [4 +- sqrt(12)]/2
---
x = [4 +- 2sqrt(3)]/2
---
x = [2 +- sqrt(3)]
x = 2+sqrt(3) or x = 2-sqrt(3) or x = -1
============================================
Cheers,
Stan H.
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Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


If -1 is a zero, then must be a factor of
Use synthetic division or polynomial long division to divide by . Too difficult to render on this site, so review the process at http://www.purplemath.com/modules/synthdiv.htm or http://www.purplemath.com/modules/polydiv2.htm

The quotient comes out to be

Use the quadratic formula to determine the roots of the quotient polynomial are . Verification is left as an exercise for the student.

Hence the remaining two factors of the original polynomial are:



and



So:



John
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!



Move the "+1" term:



Factor the first two terms, , as 
Factor the last two terms, , as 




Factor out 







Now we have to find the two zeros of 
which is not factorable with integers.  So we use the 
quadratic formula:

 


 





So the other two zeros are 

 and 

or

 and 

So now 



becomes:



Edwin
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