Question

If \(\alpha, \beta\) are the roots of the quadratic equation \(x^2 + px + q = 0\), then the values of \(\alpha^{3}, \beta^{3}\) and \(\alpha^{4}+\alpha^{2}\beta^{3}+\beta^{4}\) are respectively

• A. $3pq - p^3$ and $p^4 - 3p^2q + 3q^2$
• B. $-p(3q - p^2)$ and $(p^2 - q)(p^2 + 3q)$
• C. $pq - 4$ and $p^4 - q^4$
• D. $3pq - p^3$ and $(p^2 - q) (p^2 - 3q)$

Answer 1

The Correct Option is D

The correct answer is D:\(3pq-p^3\space and \space (p^2-q)(p^2-3q)\)
Given that;
Quadratic equation is \(x^2+px+q=0\)
\(\because Sum\space of\space roots, \alpha+\beta=-p and \alpha \beta=q\)
\(\therefore\left(\alpha^{3}+\beta^{3}\right)=(\alpha+\beta)^{3}-3 \alpha \beta(\alpha+\beta)\)
\(=(-p)^{3}-3 q(-p)\)
\(=-p^{3}+3 p q\)
\(and\space \alpha^{4}+\alpha^{2} \beta^{2}+\beta^{4}=\left(\alpha^{4}+\beta^{4}\right)+(\alpha \beta)^{2}\)
\(=\left(\alpha^{2}+\beta^{2}\right)^{2}-(\alpha \beta)^{2}\) 
\( = {\left[(\alpha+\beta)^{2}-2 \alpha \beta\right]^{2}-(\alpha \beta)^{2}}\) 
\( =\left[(-p)^{2}-2 q\right]^{2}-3(q)^{2}\) 
\( =\left[p^{2}-2 q\right]^{2}-3 q^{2}\) 
\(=p^{4}-4 p^{2} q+4 q^{2}-q^{2}\) 
\(=p^{4}-4 p^{2} q+3 q^{2}\) 
\( =p^{4}-3 p^{2} q-p^{2} q+3 q^{2}\) 
\( =p^{2}\left(p^{2}-3 q\right)-q\left(p^{2}-3 q\right)\) 
\(= \left(p^{2}-q\right)\left(p^{2}-3 q\right)\)