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SOLUTION: This is really tough. Find the solution polynomial P(x) given zeros at: i, and – 3 +√3 (assume P(x) has rational coefficients).
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-> SOLUTION: This is really tough. Find the solution polynomial P(x) given zeros at: i, and – 3 +√3 (assume P(x) has rational coefficients).
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Question 87744
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This is really tough. Find the solution polynomial P(x) given zeros at: i, and – 3 +√3 (assume P(x) has rational coefficients).
Answer by
ankor@dixie-net.com(18980)
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Find the solution polynomial
P(x) given zeros at: i, and – 3 +√3 (assume P(x) has rational coefficients).
:
x = i
x^2 = i^2; square both sides:
x^2 = -1; remember that i^2 = -1
x^2 + 1 = 0; add one to both sides,
One factor is (x^2 + 1)
:
x = -3 + Sqrt(3)
x + 3 = Sqrt(3); added 3 to both sides
(x+3)^2 = 3; squared both sides
x^2 + 6x + 9 = 3; FOILed (x+3)^2
x^2 + 6x + 9 - 3 = 0; subtract 3 both side
x^2 + 6x + 6 = 0; the 2nd factor
:
Multiply (x^2 + 6x + 6) times (x^2 + 1)
:
you should get: P(x) = x^4 + 6x^3 + 7x^2 + 6x + 6