Question

7. If a and B are the zeroes of the polynomial f(x) = 5x2 – 7x + 1, then find the
value of alpha/beta + beta/alpha. plzzz solve with full explaination and steps​

Answer 1

Answer

47

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Given

f(x) = 5x² - 7x + 1

Comparing with the form ax² + bx + c

a = 5

b = -7

c = 1

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If α and β are the zeroes of the polynomial

then,

Sum of the zeroes =

(-coefficient of x) / coefficient of x²

$\rm{\implies\: \alpha + \beta = \frac{-(b) }{a}}$

$\rm{\implies\: \alpha + \beta = \frac{-(-7) }{1}}$

$\rm{\implies\: \alpha + \beta = 7\:\rightarrow(1)}$

Product of the zeroes = constant term / coefficient of x²

$\rm{\implies\: \alpha × \beta = \frac{c }{a}}$

$\rm{\implies\: \alpha \beta = \frac{1}{1}}$

$\rm{\implies\: \alpha \beta = 1\:\rightarrow(2)}$

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Now

$\large\rm{\frac{\alpha}{\beta} + \frac{\beta}{\alpha}}$

$\large\rm{\implies\: \frac{\alpha^{2} + \beta^{2}}{\alpha\beta}}$

$\large\rm{\implies\: \frac{(\alpha + \beta)^{2} - 2\alpha \beta }{\alpha\beta}}$

Using (1) and (2)

$\large\rm{\implies\: \frac{(7)^{2} - 2(1) }{1}}$

$\large\rm{\implies\: \frac{49 - 2}{1}}$

$\large\rm{\implies\: 47}$