Question

If alpha and beta are the zeros of the o
polynomial f(x)=x^+x+1 then find the value of 1/alpha and 1/beta​

Answer 1

⇝ Appropriate Question :

If α and β are the zeros polymomial f(x) = x² + x + 1 The Find The Value of $\sf\dfrac{1}{\alpha}+\dfrac{1}{\beta}$

⇝ Given :

α and β are the zeros polymomial

f(x) = x² + x + 1.

⇝ To Find :

Value of $\sf\dfrac{1}{\alpha}+\dfrac{1}{\beta}$

⇝ Main Concept :

☆ Relationship between Zeros and coefficiants of a Quadratic Polynomial :

For a qudratic polynomial of the Form ax² + bx + c

• Sum of Zeros = $\sf-\dfrac{b}{a}$

• Product of Zeros = $\sf\dfrac{c}{a}$

⇝ Solution :

We Have ,

α and β are the zeros polymomial

f(x) = x² + x + 1.

Hence ,

$\large \pmb{ \alpha + \beta = - 1 } \: \: \:----(1) \:$

$\large\pmb{ \alpha \beta = 1} \: \: \:---- (2)$

Now,

$\large \dfrac{1}{ \alpha } + \frac{1}{ \beta }$

$\large \sf = \frac{ \beta + \alpha }{ \alpha \beta }$

★ Using (1) and (2) :

$\large\sf = \frac{ - 1}{1} \\ \\ = -1$

$\purple{ \Large :\longmapsto  \underline { \pmb{\boxed{{  \frac{1}{ \alpha } + \frac{1}{ \beta } = - 1 } }}}}$

Answer 2

Correct QuesTion

• If alpha and beta are the zeros of the of polynomial x² + x + 1 then find the value of 1/alpha + 1/beta.

Required AnsweR

• Value of 1/alpha + 1/beta is -1

Step By Step Explanation

Given that:

• Alpha and beta are the zeros of the of polynomial f(x) = x² + x + 1

To Find:

• The value of 1/alpha + 1/beta

Solution:

❏ Things to know before solving :

• $\pmb{\boxed{\bf{\purple{\alpha + \beta = \dfrac{-b}{a}}}}}$

• $\pmb{\boxed{\bf{\red{\alpha\beta = \dfrac{c}{a}}}}}$

❏ Finding value of 1/alpha + 1/beta :

⇒ 1/alpha + 1/beta = $\sf \dfrac{1}{\alpha} + \dfrac{1}{\beta}$

$\:$

⇒ 1/alpha + 1/beta = $\sf \dfrac{\beta + \alpha}{\alpha\beta}$

$\:$

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

$\:$

⇒ 1/alpha + 1/beta = $\sf \dfrac{\dfrac{-1}{1}}{\dfrac{1}{1}}$

$\:$

⇒ 1/alpha + 1/beta = $\sf \dfrac{-1}{1}\:\times\:\dfrac{1}{1}$

$\:$

⇒ 1/alpha + 1/beta = $\sf \dfrac{-1\:\times\:1}{1\:\times\:1}$

$\:$

⇒ 1/alpha + 1/beta = $\sf \dfrac{-1}{1}$

$\:$

➦ 1/alpha + 1/beta = $\pmb{\underline{\boxed{\bf{\pink{-1}}}}}$

∴ Hence, value of 1/alpha + 1/beta is -1.

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