Question

if alpha and beta are zeroes of x^2-px+q find ;

alpha^3+beta^3

​1/alpha^3+1/beta^3

alpha^2/beta+beta^2/alpha

Answer 1

Answer.

Let,

$f(x) = {x}^{2} - px + q$

Since, α and β are zeros of f(x)

So,

$\alpha + \beta = -( \frac{ - p}{1} ) = p$

And,

$\alpha \beta = \frac{q}{1} = q$

1.

${ \alpha }^{3} + { \beta }^{3}$

$= ({ \alpha + \beta })^{3} - 3 \alpha \beta ( \alpha + \beta )$

$= {p}^{3} - 3pq \: \: ...(1)$

2.

$\frac{1}{ { \alpha }^{3} } + \frac{1}{ { \beta }^{3} }$

$= \frac{ { \alpha }^{3} + { \beta }^{3} }{ { \alpha \beta }^{3} }$

$\frac{ {p}^{3} - 3pq}{ {q}^{3} } \: \: \: \: \: \: \: \: \: (from \: (1))$

3.

$\frac{ { \alpha }^{2} }{ \beta } + \frac{ { \beta }^{2} }{ \alpha }$

$= \frac{ { \alpha }^{3} + { \beta }^{3} }{ \alpha \beta }$

$\frac{ {p}^{3} - 3pq }{q} \: \: \: \: \: \: \: \: \: \: \: \: \: \: (from \: (1))$

Please mark the BRAINLIEST.

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