Question

If $\alpha ,\beta$ are the zeros of the polynomials $f\left(x\right)=x^2+x+1$ then $\frac{1}{\alpha }+\frac{1}{\beta }=$________.

• A. $1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

Answer: (C)

$-1$

$f\left(x\right)=x^2+x+1$

$a=1$

$b=1$

$c=1$

$\because \alpha$ and $\beta$ are the zeroes of above polynomial.

$\therefore$ Sum of roots $=\frac{-b}{a}$

$\Rightarrow \alpha +\beta =\frac{-1}{1}$

$\Rightarrow \alpha +\beta =-1⟶\left(1\right)$

Product of roots $=\frac{c}{a}$

$\Rightarrow \alpha \beta =\frac{1}{1}$

$\Rightarrow \alpha \beta =1⟶\left(2\right)$

$\therefore \frac{1}{\alpha }+\frac{1}{\beta }=\frac{\beta +\alpha }{\alpha \beta }$

From ${eq}^n\left(1\right)\&\left(2\right)$, we have

$\Rightarrow \frac{1}{\alpha }+\frac{1}{\beta }=\frac{-1}{1}=-1$

Hence, -1 is the correct answer.