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Relationship Between Zeroes and Coefficients of a Quadratic Polynomial

If α, β are...

Question

If $α, β$ are zeroes of the polynomial $p (x) = x^{2} - x - 4$ , then find the value of $\frac{α}{β} + \frac{β}{α} + 3 (\frac{1}{α} + \frac{1}{β}) - 2 α β$ ..

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Solution

Given, $α, β$ are zeroes of the polynomial $p (x) = x^{2} - x - 4$ , Then from the relation between the relation roots and co-efficients we get,
$α + β = 1$ .....(1) and $α . β = - 4$ ......(2).
Now,
$\frac{α}{β} + \frac{β}{α} + 3 (\frac{1}{α} + \frac{1}{β}) - 2 α β$
$= \frac{α^{2} + β^{2}}{α . β} + 3 \frac{α + β}{α . β} - 2 α . β$
$= \frac{(α + β)^{2} - 2 α . β}{α . β} + 3 \frac{α + β}{α . β} - 2 α . β$
$= - \frac{9}{4} - \frac{3}{4} + 8$ [ Using (1) and (2)]
$= - 3 + 8$
$= 5$ .

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