Question

If p and q are the zeros of the polynomial f(x)= x^2 -5x +1, then find the value of (p^2)q + p(q^2).​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:p \: and \: q \: are \: zeroes \: of \: the \: polynomial \: {x}^{2} - 5x + 1$

We know,

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\bf\implies \:pq = \dfrac{1}{1} = 1$

And

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\bf\implies \:p + q = - \: \dfrac{( - 5)}{1} = 5$

Now, Consider

$\rm :\longmapsto\: {p}^{2}q + {qp}^{2}$

$\rm \:  =  \:pq(p + q)$

$\rm \:  =  \:1 \times 5$

$\rm \:  =  \:5$

Hence,

$\rm :\longmapsto\: \boxed{ \bf \:{p}^{2}q + {qp}^{2} = 5}$

Additional Information :-

$\red{\rm :\longmapsto\: \alpha , \beta , \gamma \: are \: zeroes \: of \: a {x}^{3} + b {x}^{2} + cx + d, \: then}$

$\boxed{ \bf{ \: \alpha + \beta + \gamma = - \dfrac{b}{a}}}$

$\boxed{ \bf{ \: \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}}}$

$\boxed{ \bf{ \: \alpha \beta \gamma = - \dfrac{d}{a}}}$