Question

If alpha and beta are the zeroes of the polynomial x^2+5x-66, then the quadratic polynomial whose zeroes are 1/alpha and 1/beta can be equal to:
1: 5x^2+66x+1
2: 5x^2-66x -1
3:66x^2-5x-1
4:66x^2+12x-1

Which one is correct??


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Answer 1

Answer: The correct option 66x^2-5x-1.

Step-by-step explanation:

As if alpha and beta are the zeroes of the polynomial x²+5x-66,

Then the quadratic polynomial whose zeroes are 1/alpha and 1/beta can be equal to:

As x²+5x-662

x = $\frac{-(5) \sqrt{5^{2}-4 \times 66 \times 1 } )}{2}$

x = $\frac{-5 \sqrt{25 + 264} }{2}$

x = $\frac{-5 \sqrt{289} }{2}$

As α = $\frac{-5+17}{2}$        β = $\frac{-5-17}{2}$

    α = $\frac{12}{2} =6$        β = -11

As $\frac{1}{\alpha } = \frac{1}{6} , \frac{1}{\beta } = \frac{-1}{11}$

So now by a quadratic polynomial equation which has had root $\frac{1}{\alpha } , \frac{1}{\beta }$

$x^{2}-(\frac{1}{6}-\frac{1}{11})x+(\frac{1}{6}\times (\frac{-1}{11}))x^{2}- \frac{(11-6)x}{66}-\frac{1}{66}\\\frac{66x^{^{2}}-5x-1}{66}\\x^{2}-(\tfrac{5}{66}x)-\frac{1}{66}$'

Hence, C is the correct option 66x^2-5x-1.

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