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Question
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find a quadratic polynomial whose zeroes are `(2alpha)/beta" and "(2beta)/alpha`
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Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 − 1
The roots are α and β
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`alpha+beta=0/1`
`alpha+beta=0`
`alphabeta="constant term"/("coefficient of "x^2)`
`alphabeta=(-1)/1`
`alphabeta=-1`
Let S and P denote respectively the sum and product of zeros of the required polynomial. Then,
`S=(2alpha)/beta+(2beta)/alpha`
Taking least common factor we get,
`S=(2alpha^2+2beta^2)/(alphabeta)`
`S=(2(alpha^2+beta^2))/(alphabeta)`
`S=(2[(alpha+beta)-2alphabeta])/(alphabeta)`
`S=(2[(0)-2(-1)])/-1`
`S=(2[-2(-1)])/-1`
`S=(2xx2)/-1`
`S=4/-1`
S = -4
`P=(2alpha)/betaxx(2beta)/alpha`
P = 4
Hence, the required polynomial f(x) is given by,
f(x) = k(x2 - Sx + P)
f(x) = k(x2 -(-4)x + 4)
f(x) = k(x2 +4x +4)
Hence, required equation is f(x) = k(x2 +4x +4) Where k is any non zero real number.
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