If α and β are the zeros of the polynomial f(x) = 5x2+4x−9 then evaluate the following: ∝3-β3

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If α and β are the zeros of the polynomial f(x) = 5 $x^{2}$ +4x−9 then evaluate the following: $\propto^{3} - β^{3}$

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\frac{874}{125}

\frac{814}{125}

\frac{894}{125}

\frac{854}{125}

answer is D.

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Detailed Solution

The given polynomial is f(x)= 5

x^{2}

+4x−9 which is a quadratic equation therefore it has two roots α and β.
Now, if a

x^{2}

+bx +c is a quadratic equation and it has two roots α and β then we can write α+β=−

\frac{b}{a}

and αβ=

\frac{c}{a}

. Therefore, for the quadratic equation
f(x)= 5

x^{2}

+4x−9 we can write α+β=−

\frac{4}{5}

and αβ=−

\frac{9}{5}

.
We know that

{(a - b)}^{3}

a^{3} - b^{3}

−3ab(a−b) then we can write

{(α - β)}^{3}

α^{3} - β^{3}

−3αβ(α−β)
⇒

α^{3} - β^{3}

{(α - β)}^{3}

+3αβ(α−β)
We have to find the value of (α−β)
Now, we know that

{(α + β)}^{2}

α^{2} + β^{2}

+2αβ
⇒

α^{2} + β^{2}

{(α + β)}^{2}

−2αβ
⇒

α^{2} + β^{2}

{(- \frac{4}{5})}^{2}

−2(

\frac{- 9}{5}

)
Now, simplify the above equation
⇒

α^{2} + β^{2}

\frac{16}{25}

\frac{18}{5}

\frac{106}{25}

Now, we know that

{(α - β)}^{2}

α^{2} + β^{2}

-2αβ . Substitute the value in the equation
⇒

{(α - β)}^{2}

\frac{106}{25}

−2(

\frac{- 9}{5}

\frac{106}{25}

\frac{18}{5}

⇒

{(α - β)}^{2}

\frac{196}{25}

Hence, from the above calculation we got the value of α−β=

\sqrt \frac{196}{25}

\frac{14}{5}

Now, substitute the value of α−β=

\frac{14}{5}

in the equation

α^{3} - β^{3}

{(α - β)}^{3}

+3αβ(α−β)
⇒

α^{3} - β^{3}

{(\frac{14}{5})}^{3}

+3(

\frac{- 9}{5}

)(

\frac{14}{5}

)
⇒

α^{3} - β^{3}

\frac{2744}{125}

−

\frac{378}{25}

\frac{2744 - 1890}{125}

Therefore, we got

α^{3} - β^{3} = \frac{854}{125}

Hence, the correct option is (4).

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