Question

If alpha, Beta, gamma are the zeroes of the
cubic polynomial p of x = x^2- 5x-1
Then find the value of alpha beta square + alpha square beta​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{\alpha\beta^{2}+\alpha^{2}\beta=-5}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given: }} \\ \tt: \implies p(x) = {x}^{2} - 5x - 1 \\ \\ \tt: \implies \alpha \: and \: \beta \: are \: the \: zeroes\\ \\ \red{\underline \bold{To \: Find: }} \\ \tt: \implies \alpha { \beta }^{2} + { \alpha }^{2} \beta = ?$

• According to given question :

$\tt \circ \: {x}^{2} - 5x - 1 = 0 \\ \\ \tt \circ \: a = 1 \\ \\ \tt \circ \: b = - 5 \\ \\ \tt \circ \: c = - 1 \\ \\ \bold{For \: Sum \: of \: zeroes} \\ \tt: \implies \alpha + \beta = \frac{ - b}{a} \\ \\ \tt: \implies \alpha + \beta = \frac{ - ( - 5)}{1} \\ \\ \green{\tt: \implies \alpha + \beta =5} \\ \\ \bold{For \: Product \: of \: zeroes} \\ \tt: \implies \alpha\beta = \frac{c}{a} \\ \\ \tt: \implies \alpha \beta = \frac{ - 1}{1} \\ \\ \green{\tt: \implies \alpha \beta = - 1} \\ \\ \bold{For \: finding \: zeroes} \\ \tt: \implies \alpha { \beta }^{2} + { \alpha }^{2} \beta \\ \\ \tt: \implies \alpha \beta ( \beta + \alpha ) \\ \\ \tt: \implies - 1(5) \\ \\ \green{\tt: \implies - 5} \\ \\ \green{\tt \therefore \alpha { \beta }^{2} + { \alpha }^{2} \beta = - 5}$

Answer 2

AnswEr :

Explanation :

The given polynomial is p(x) = x² - 5x - 1.

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

NoTE

The relations between the zeros and the coefficients for a quadratic polynomial are given as :

Consider a polynomial ax² + bx + c

• Sum of Zeros : - b/a

• Product of Zeros : c/a

Here,

Sum of Zeros

$\sf \: \alpha + \beta = - ( - 5) = 5$

Product of Zeros

$\sf \: \alpha \beta = - 1$

Now,

$\alpha \beta {}^{2} + { \alpha }^{2} \beta \\ \\ \longrightarrow \: \alpha \beta ( \alpha + \beta ) \\ \\ \longrightarrow \: \sf \: ( - 1)( 5) \\ \\ \longrightarrow \sf \: - 5$

Thus,the value of $\alpha \beta^2 + \alpha^2 \beta$ is - 5