Question

If \( \alpha, \beta, \gamma \) are the roots of the quadratic equation \( x^{3}-p x^{2}+q x-r=0 \), then the value of \( \sum \alpha^{2} \beta \) is equal

Answer 1

α β and γ are roots of cubic equation x³ - px² + qx - r = 0

\(\therefore\) sum of roots = \(\cfrac{-(-p)}1\) = p

\(\therefore\) α + β + γ = p....(1)

sum of product of two roots = q/1 = q

\(\therefore\) αβ + βγ + γα = q....(2)

products of roots = \(\cfrac{-(-r)}1\) = r

\(\therefore\) αβγ = r....(3)

multiplying (1) and (2) we get

pq = ( α + β + γ) (αβ + βγ + γα)

= pq = α²β + αβγ + α²γ + αβ²+ β²γ + αβγ + αβγ + βγ² +γ²α

= (α²β + β²γ + α²γ + α²γ + γ²β + β²α) + 3αβγ

\(\sum\)= α²β + 3r

(\(\because\) r = αβγ)

\(\sum\) α²β = pq -3r