1) Check that for the roots obtained in the example problem:
a) (a1 x a2) +
(a
2 x a3)
+ (a3
x a1)
= q
AND
b) a1 x a2 x a3 = r
AND
b) a1 x a2 x a3 = r
Solution: From the example problem we saw:
a1 =
1.103 - 0.665i
a2 = 1.103 + 0.665i
a3 = -1.206
a2 = 1.103 + 0.665i
a3 = -1.206
Then: (a1 x a2) = (1.103 - 0.665i) (1.103 + 0.665i ) = 1.659
(a
2 x a3) =
(1.103 + 0.665i) (-1.206) = -1.33 – 0.802i
(a3
x a1)
= (-1.206) (1.103 - 0.665i = -1.33 + 0.802i
So: (a1 x a2) +
(a
2 x a3)
+ (a3
x a1)
=
1.659 + (-1.33 – 0.802i) + (-1.33 + 0.802i) » 1.66 – 2.66 =
-1 = q
b) a1 x a2 x a3 = (1.103 - 0.665i) (1.103 + 0.665i)( -1.206) » - 2 = r
2) Solve: x3 – 6x2 + 11x – 6
Then: p = 6, q = -11, r = 6
r 1= - 1/2 + 1/2 Ö- 3
r 2= - 1/2 - 1/2 Ö- 3
So: D = 60.033i
x1 = [p3 –9/2 (pq – 3r) – 3/2 x (-3)1/2 x D ]1/3
= 9.085
x2 = (p2 – 3q)/ x1
= 7.595
The roots are then:
a1 = 1/3(p + r2 x x1 + r x x2) = 7.56
a2 = 1/3( p + r2 x x1+ r x x2) = -0.78 – 0.43i
a3 = 1/3( p + r x x1 + rx r2 x x2)
= -0.78 + 0.43i
Check: a1 + a2 + a3 = p = 6
= 7.56 + (-0.78 – 0.43i) + ( -0.78 + 0.43i) = 7.56 - 1.56 = 6
(a1
x a2)
+ (a
2 x a3)
+ (a3
x a1)
= q = -11
(a1
– a2)
x (a2
– a3) x
(a3
– a1)
= D
= 60.033i
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