- #1

rock.freak667

Homework Helper

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## Main Question or Discussion Point

Considering the roots of a cubic polynomial([itex]ax^3+bx^2+cx+d[/itex]),[itex]\alpha,\beta,\gamma[/itex]

[tex]\sum \alpha=\frac{-b}{a}[/tex]

[tex]\sum \alpha\beta=\frac{c}{a}[/tex]

[tex]\sum \alpha\beta\gamma=\frac{-d}{a}[/tex]

If I have those sums of roots..and I am told to find [itex]\alpha^9+\beta^9+\gamma^9[/tex] is there any easy way to find this without having to expand?

and also for a quartic polynomial

when I expand [itex](x-\alpha)(x-\beta)(x-\gamma)(x-\delta)[/itex]

I get:

[tex]x^4-(\alpha+\beta+\alpha\gamma+\beta\gamma+\alpha\delta+\beta\gamma)x^3+(\alpha\beta+\gamma\delta+\alpha\gamma+\beta\gamma+\alpha\delta+\beta\delta)x^2 -(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\gamma\delta\beta)x+\alpha\beta\gamma\delta[/tex]

for -x^3 I am supposed to get the sum of the roots...yet I expanded correctly, where did i go wrong?

[tex]\sum \alpha=\frac{-b}{a}[/tex]

[tex]\sum \alpha\beta=\frac{c}{a}[/tex]

[tex]\sum \alpha\beta\gamma=\frac{-d}{a}[/tex]

If I have those sums of roots..and I am told to find [itex]\alpha^9+\beta^9+\gamma^9[/tex] is there any easy way to find this without having to expand?

and also for a quartic polynomial

when I expand [itex](x-\alpha)(x-\beta)(x-\gamma)(x-\delta)[/itex]

I get:

[tex]x^4-(\alpha+\beta+\alpha\gamma+\beta\gamma+\alpha\delta+\beta\gamma)x^3+(\alpha\beta+\gamma\delta+\alpha\gamma+\beta\gamma+\alpha\delta+\beta\delta)x^2 -(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\gamma\delta\beta)x+\alpha\beta\gamma\delta[/tex]

for -x^3 I am supposed to get the sum of the roots...yet I expanded correctly, where did i go wrong?