Question

if alpha and beta are the zeroes of the polynomial f(x) = x^2 - px + q then find the value of alpha/beta + beta/alpha

Answer 1

Answer:

We have been given that α and β are the zeroes of polynomial x² + px + q. Therefore, Required Value of (α /ß + 2)(ß/α + 2) is 2p² + q.

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Answer 2

Answer:

The required value is  $\tt \dfrac{p^2}{q}-2$

Step-by-step explanation:

f(x) = x² - px + q

α and β are the zeroes of the polynomial.

Relation between zeroes and the coefficients of quadratic polynomial :

⇒ Sum of zeroes = -(x coefficient)/x² coefficient

  α + β = -(-p)/1

 α + β = p

⇒ Product of zeroes = constant/x² coefficient

  α × β = q/1

 αβ = q

We know,

 (x + y)² = x² + y² + 2xy

Similarly,

(α + β)² = α² + β² + 2αβ

  p² = α² + β² + 2q

 α² + β² = p² - 2q

We have to find the value of α/β + β/α

  $\longrightarrow \sf \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha} \\\\ \implies \sf \dfrac{\alpha ^2+\beta ^2}{\alpha \beta} \\\\ \implies \sf \dfrac{p^2-2q}{q} \\\\ \implies \sf \dfrac{p^2}{q}-\dfrac{2q}{q} \\\\ \implies \sf \dfrac{p^2}{q}-2$

Therefore, the required value is  $\rm \dfrac{p^2}{q}-2$