Question 192382: Find exact values for all real/complex number solutions to the polynomial equation...
3x^4 - 5x^3 + 5x^2 - 5x + 2 = 0
Could you please provide me with a very detailed explanation of how you achieved the solution. Thanks!
Answer by RAY100(1637)   (Show Source):
You can put this solution on YOUR website! techniques that apply include: rational zero theorem, synthetic division, and fundamental theorem of algebra.
Fundamental theorem states that a 4th degree function has 4 roots.
rational zero theorem states that zeros are ratios of factors of last term coeff and leading coeff
coeff are 2,3 therefore factors are +/- ( 2/1, 1/3, 1/1, 2/3)
using synthetic div first
1 3 -5 5 -5 2
3 -2 3 -2
_________________
3 -2 3 -2 0
therefore (x-1) is a factor, x=1 is ZERO
remaining poly is 3x^3-2x^2+3x-2
try factoring by grouping, rearranging,
3x^3 +3x -2x^2 -2
3x(x^2+1) -2(x^2+1)
(3x-2) (x^2+1)
setting 1st term =0
3x-2=0
3x=2
x=2/3 ZERO
2nd term, (x^2+1) is not readily factored but quadratic eqn works
using a=1, b=0, c=1
x= ( -0 +/- sq rt ( 0^2 -(4) (1) (1) ) / 2 *1
x= ( +/- sq rt ( -4 ) ) /2
x=+/- 2i/2=+/- (i), LAST 2 ZEROS
where i= sq rt (-1)
in summary, zeros are (1), (2/3), (+i), (-i)
check by multiplying factors
(x^2+1) (x-1) (3x-2)
(x^2+1) (3x^2 -5x+2)
3x^4 -5x^3 +2x^2 +3x^2 -5x+2
3x^4-5x^3+5x^2-5x+2 ok
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