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Integration by Partial Fractions

If α-β=√b2-...

Question

If $(α - β) = \frac{\sqrt{b^{2} - 4 a c}}{a}$ and $α β = \frac{c}{a}$ then find $α^{3} - β^{3}$

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Solution

$α^{3} - β^{3} = (α - β) (α^{2} + α β + β^{2})$
$= (α - β) (α^{2} + β^{2} + α β)$
We have ${(α - β)}^{2} = α^{2} + β^{2} - 2 α β$
or $α^{2} + β^{2} = {(α - β)}^{2} + 2 α β$
$α^{3} - β^{3} = (α - β) ({(α - β)}^{2} + 2 α β + α β)$
$= (α - β) ({(α - β)}^{2} + 3 α β)$
$= \frac{\sqrt{b^{2} - 4 a c}}{a} ⎛ ⎜ ⎝ {(\frac{\sqrt{b^{2} - 4 a c}}{a})}^{2} + \frac{3 c}{a} ⎞ ⎟ ⎠$
$= \frac{\sqrt{b^{2} - 4 a c}}{a} (\frac{b^{2} - 4 a c}{a^{2}} + \frac{3 c}{a})$
$= \frac{\sqrt{b^{2} - 4 a c}}{a} (\frac{b^{2} - 4 a c + 3 c a}{a^{2}})$
$= \frac{\sqrt{b^{2} - 4 a c}}{a} (\frac{b^{2} - a c}{a^{2}})$
$= \frac{(\sqrt{b^{2} - 4 a c}) (b^{2} - a c)}{a^{3}}$

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