Question

If α and β are the zeroes of the Quadratic Polynomials p(x) = 4x2 - 5x – 1. Find the value of α2β + αβ2
i) 5/16 ii) 3/16 iii) -5/16 iv) none

Answer 1

        $\text{\huge \bf \underline{Answer:}}$

        $\text{It is given that }\alpha \text{ and }\beta \text{ are roots of the polynomial}\\p(x) = 4x^{2} - 5x - 1$

        $\text{We have to find the value of }\alpha^{2}\beta + \alpha\beta^{2}$

        $\text{Taking }\alpha\beta \text{ common}$

$\implies\: \alpha\beta(\alpha + \beta)$

        $\text{Applying the relationship between roots and coefficients}$

$\implies\: \text{Sum of roots} \longrightarrow \alpha + \beta = \dfrac{-b}{a} \longrightarrow \alpha + \beta = \dfrac{-(-5)}{4} \longrightarrow \alpha + \beta = \dfrac{5}{4}$

$\implies \:\text{Product of roots} \longrightarrow \alpha\beta = \dfrac{c}{a} \longrightarrow \alpha\beta = \dfrac{-1}{4} \longrightarrow \alpha\beta =- \dfrac14$

        $\text{Now, substituting the values in the expression}$

$\implies \:\alpha\beta(\alpha + \beta) = -\dfrac14\text{\huge{(}}\dfrac54\text{\huge{)}}$

$\implies\: \alpha\beta(\alpha + \beta) = -\dfrac{5}{16}$

$\implies \:\boxed{\boxed{\alpha^{2}\beta + \alpha\beta^{2} = -\dfrac{5}{16}}}$