Question

Hey Mates!

If $\alpha \beta$
are the roots of $a {x}^{2} + bx + c = 0$
then
${ \alpha }^{3} { \beta }^{3} + { \alpha }^{2} { \beta }^{2} + { \alpha }^{3} { \beta }^{3}$
is equal to ?

No silly answers.

Answer 1

Hola there,

Let alpha be 'A' and beta be 'B'.

So, now according to the question we have A and B as zeroes of polynomial ax² + bx + c = 0

So,
As we know,

A + B = -b/a
AB = c/a

Now, we have to solve..

=> A³B³ + A²B² + A³B³

=> 2A³B³ + A²B²

=> A²B²(2AB + 1)

=> (AB)²(2AB + 1)

=> c²/a²(2c/a + 1)

=> c²/a²[(2c + a)/a]

=> c²(2c + a)/a³. ...Ans

Hope this helps....:)

Answer 2

hi friend
here is your answer
$\alpha \beta = \frac{c}{a} \\ { \alpha }^{3} { \beta }^{3} + { \alpha }^{2} { \beta }^{2} + { \alpha }^{3} { \beta }^{3} \\ = ( { \frac{c}{a} })^{3} + ( { \frac{c}{a} })^{2} + ({ \frac{c}{a} })^{3} \\ = \frac{ {c}^{3} }{ {a}^{3} } + \frac{ {c}^{2} }{ {a}^{2} } + \frac{ {c}^{3} }{ {a}^{3} } \\ = \frac{2 {c}^{3} + a {c}^{2} }{ {a}^{3} }$