Question

If ,ß are the roots of ax² +bx+c = 0,
then α ‐³ + ß‐³ / α³ + ß³ is ​

Answer 1

If ,ß are the roots of ax² +bx+c = 0,

then α ‐³ + ß‐³ / α³ + ß³ is 0

Answer 2

${\maltese \; \;{\underline{\purple{\underline{\textsf{\textbf{Given that :}}}}}}}$

★ Quadratic polynomial : ax² + bx + c = 0

${\maltese \; \;{\underline{\purple{\underline{\textsf{\textbf{To Find :}}}}}}}$

★ The value of thee expression : α⁻³ + β ⁻³ / α³ + β³

${\maltese \; \;{\underline{\purple{\underline{\textsf{\textbf{Assuming that :}}}}}}}$

★ α and β are said to be thee roots of the quadratic polynomial

${\maltese \; \;{\underline{\purple{\underline{\textsf{\textbf{We know that :}}}}}}}$

★ The sum of the roots alpha ( α ) and beta ( β ) = - b / a

★ The product of the roots α and β equals to c / a

${\maltese \; \;{\underline{\purple{\underline{\textsf{\textbf{Full Solition :}}}}}}}$

Now, we've been said that a quadratic polynomial which could be further classified into a trinomial has 2 roots which are α and β and asked to find out the value of the expression :

$\bigstar \; {\underline{\boxed{\tt{ \dfrac{\alpha ^{-3} + \beta^{-3}}{\alpha^3 + \beta^3} }}}}$

Since we have 2 formulas which state the relation betwen the roots and the constant terms in the equation ax² + bx + c = 0 let's expand the required expression into such a form that we can apply the formulas we have and find the value of :

$: {\implies } \sf \dfrac{\alpha ^{-3} + \beta^{-3}}{\alpha^3 + \beta^3}$

Now the laws used here state that :

• α + β = - b / a
• αβ = c / a

Now let's expand the required expression such that the above mentioned laws can be applied,

$: {\implies } \sf \dfrac{\alpha ^{-3} + \beta^{-3}}{\alpha^3 + \beta^3}$

$: {\implies } \sf \dfrac{\dfrac{1}{\alpha^3} + \dfrac{1}{\beta^3} }{\alpha^3 + \beta^3}$

$: {\implies } \sf \dfrac{1}{\alpha^3} + \dfrac{1}{\beta^3} \times \dfrac{1}{\alpha^3 + \beta^3}$

$: {\implies } \sf \dfrac{\cancel{\beta^3 + \alpha^3}}{\alpha^3\beta^3} \times \dfrac{1}{\cancel{\alpha^3 + \beta^3}}$

$: {\implies } \sf \dfrac{1}{\alpha^3 \beta^3}$

$: {\implies } \sf \dfrac{1}{(\alpha \beta )^3}$

Since the product of the roots equals c / a ,

$: {\implies } \sf \dfrac{1}{\bigg\{\dfrac{c}{a} \bigg\}^3}$

Henceforth the problem is Solved.