Question

##CHALLENGE FOR EVERYONE##
*LET'S SEE WHO CAN SOLVE THIS QUESTION*

Q.
$if \: \alpha \: and \: \beta are \: roots \: of \: the \: equation \: {x}^{2} + 5x + 3 = 0 \: \\ then \: find \: the \: value \: of \: \frac{1}{ { \alpha }^{3} } - \frac{1}{ { \beta }^{3} }$
*NO USELESS ANS. PLZ.*

Answer 1

p(x) = x² + 5x + 3
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Sum of roots = -b/a

            α + β = -(5)/1

            α + β = -5
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Product of zeroes = c/a

                        αβ = 3/1

                        αβ = 3
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Then 1/α³ - 1/β³ equal to

= 1/α³ - 1/β³

= β³ - α³ / (αβ)³

= (β+α)(α²+β²-αβ) / (αβ)³ 

(as a³-b³ = (a+b)(a²+b²-ab))

= (α+β)((α+β)²-3αβ) / (αβ)³

(as (a+b)² = a²+b²+2ab)

Substituting these values,

= (-5)((-5)²-3(3)) / (3)³

= -5(25-9) / 27

= -5(16) / 27

= -80/27
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