Question

If α, β, γ be the zeros of the polynomial p(x) such that (α + β + γ) = 3, (αβ + βγ + γα) = −
10 and αβγ = −24, then p(x) = ?

(a) x³ + 3x² − 10x + 24
(b) x³ + 3x² + 10x −24
(c) x³ − 3x² −10x + 24
(d) None of these

Answer 1

$$\begin{gathered} \left(\text{c}\right)x^3-3x^2-10x+24 \\ \text{Given:}\mathrm{\alpha }\text{,}\mathrm{\beta }\mathrm{and}\mathrm{\gamma }\text{are the zeroes of polynomial}p\left(x\right). \\ \mathrm{Also},\text{(}\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })=3,\text{(}\mathrm{\alpha \beta }\text{+}\mathrm{\beta \gamma }+\mathrm{\gamma \alpha })=-10\mathrm{and}\alpha \beta \gamma \text{=}-24 \\ \therefore p\left(x\right)=x^3-\text{(}\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })x^2+\text{(}\mathrm{\alpha \beta }\text{+}\mathrm{\beta \gamma }+\mathrm{\gamma \alpha })x-\mathrm{\alpha \beta \gamma } \\ =x^3-3x^2-10x+24 \end{gathered}$$