Question

if alpha into bitaa are the zero of the polynomial f(x )a x2+ bx+c than find the valu of 1/alpha+1/ bita​

Answer 1

Answer :

The required value is -b/c

Given :

The quadratic polynomial is

• f(x) = ax² + bx + c

To Find :

The Value of

$\sf \bullet \: \: \dfrac{1}{\alpha} + \dfrac{1}{\beta}$

Concept to be used :

The relationship between the zeroes and the coefficients of the polynomial :

$\sf \bullet \: \: Sum \: \: of \: \: the \: \: zeroes =- \dfrac{coefficient \: \: of \: \: x}{coefficient \: \: of \: \: x^{2} }$

$\sf \bullet \: \: Product \: \: of \: \: the \: \: zeroes = \dfrac{constant \: \: term}{coefficient \: \: of \: \: x^{2}}$

Solution :

The given polynomial is :

ax² + bx + c

Sum of the zeroes :

$\sf \alpha + \beta = -\dfrac{b}{a} ...........(1)$

Now product of the zeroes :

$\sf \alpha\beta = \dfrac{c}{a} ..........(2)$

Dividing (1) by (2) we have :

$\sf \dfrac{\alpha}{\alpha\beta} + \dfrac{\beta}{\alpha\beta}=\dfrac{-\dfrac{b}{a}}{\dfrac{c}{a}} \\\\ \sf \implies \dfrac{1}{\beta} + \dfrac{1}{\alpha} = -\dfrac{b}{c} \\\\ \sf \implies \dfrac{1}{\alpha} + \dfrac{1}{\beta} = -\dfrac{b}{c}$

Answer 2

Answer:

⋆ Given Polynomial : ax² + bx + c

$\underline{\bigstar\:\textbf{According to the Question :}}$

$:\implies\sf \dfrac{1}{ \alpha } + \dfrac{1}{ \beta }\\\\\\:\implies\sf \dfrac{\beta+ \alpha }{\alpha \beta }\\\\\\:\implies\sf\dfrac{Sum\:of\: Zeroes}{Product\:of\: Zeroes}\\\\\\:\implies\sf \dfrac{\quad-\frac{coefficient\: \: of \: \: x}{coefficient \: \: of \: \: x^2}\quad}{\frac{constant \: \: term}{coefficient \: \: of \: \: x^2}}\\\\\\:\implies\sf \dfrac{\:\:\frac{ - \:b}{a}\:\:}{ \frac{c}{a} }\\\\\\:\implies\sf \dfrac{ - \:b}{a} \times \dfrac{a}{c}\\\\\\:\implies\underline{\boxed{\dfrac{\textsf{\textbf{- b}}}{\textsf{\textbf{c}}}}}$