Question

If alpha and beta are the zeroes of the polynomial x²+x+1 then find the value of
$\frac{ 1}{ \alpha } + \frac{1}{ \beta }$
​

Answer 1

$\huge\purple{\boxed{\underline{ANSWER}}}$

GIVEN

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

FORMULA

• In a quadratic polynomial such as ax²+bx+c ( standard form of quadratic polynomial) if two zeroes of this polynomial α and β then

$\rarr \: \: \alpha + \beta = \frac{ - b}{a}$

$\rarr \: \: \alpha \beta = \frac{c}{a}$

SOLUTION

• Given polynomial = x²+x+1

• This is a quadratic polynomial

• let compare this polynomial with standard form of quadratic polynomial ax²+bx+c , we get

a = 1 , b = 1 , c = 1

• The zeroes of this polynomial are α and β.

Now

$\rarr \: \: \alpha + \beta = \frac{ - 1}{1} = - 1$

$\rarr \: \: \alpha \beta = \frac{1}{1} = 1$

Now

$\rarr \: \: \: \: \: \: \frac{1}{ \alpha } + \frac{1}{ \beta } = \frac{ \beta + \alpha }{ \alpha \beta } \\ \rarr \: \: \frac{ \alpha + \beta }{ \alpha \beta } = \frac{ - 1}{1} = - 1$