Question

If alpha, bita are zeroes of polynomial x2+x+1, then find 1/apha+1/bita =?

Answer 1

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

$\green{\tt{\therefore{\frac{1}{\alpha}+\frac{1}{\beta}=-1}}}$

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

$\green{\underline \bold{Given :}} \\ \tt: \implies {x}^{2} + x + 1 = 0 \\ \\ \tt: \implies \alpha \: and \: \beta \: are \: zeroes \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies \frac{1}{ \alpha } + \frac{1}{ \beta } = ?$

• According to given question :

$\tt \circ \: {x}^{2} + x + 1 = 0 \\ \\ \tt \circ \: a = 1 \: \: \: \: \: \: \: b = 1 \: \: \: \: \: \: \: c = 1\\ \\ \bold{For \: sum \: of \: zeroes} \\ \tt: \implies \alpha + \beta = - \frac{b}{a} \\ \\ \tt: \implies \alpha + \beta = \frac{ - 1}{1} \\ \\ \green{\tt: \implies \alpha + \beta = - 1 }\\ \\ \bold{For \: product \: of \: zeroes} \\ \tt: \implies \alpha \beta = \frac{c}{a} \\ \\ \tt: \implies \alpha \beta = \frac{1}{1} \\ \\ \green{\tt: \implies \alpha \beta =1} \\ \\ \bold{For \: finding \: value} \\ \tt: \implies \frac{1}{ \alpha } + \frac{1}{ \beta } \\ \\ \tt: \implies \frac{ \beta + \alpha }{ \alpha \beta } \\ \\ \tt: \implies \frac{ - 1}{1} \\ \\ \green{\tt: \implies - 1} \\ \\ \green{\tt \therefore \frac{1}{ \alpha } + \frac{1}{ \beta } = - 1}$

Answer 2

Given :-

α and β are the zeros of the polynomial x² + x + 1.

To find :-

$\displaystyle{\sf{\frac{1}{\alpha } +\frac{1}{\beta } }}$

Solution :-

We know, the sum of the zeros of a quadratic polynomial :

α + β = -b/a

⇒α + β = -1/1 = -1

And, the product of the zeros of a quadratic polynomial :

αβ = c/a

⇒αβ = 1/1 = 1

Now,

$\displaystyle{\sf{\frac{1}{\alpha } +\frac{1}{\beta } }}\\\\\\\displaystyle{\sf{=\frac{\beta +\alpha }{\alpha \beta } }}\\\\\\\displaystyle{\sf{=\frac{-1}{1} }}$

Thus, $\boxed{\displaystyle{\sf{\frac{1}{\alpha } +\frac{1}{\beta }=-1 }}}$.