Question

If alpha and beta are the two zeroes of the polynomial 6x^2 − 7x+ 2, find a quadratic polynomial whose zeroes are 1/alpha and 1/beta.

Answer 1

21x²-6x+7=O is the equation

Answer 2

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p(x) = 6x^2 - 7x + 2
= 6x^2 - 3x - 4x + 2
= 3x ( 2x - 1 ) -2 ( 2x - 1 )
= ( 2x - 1 ) ( 3x - 2 )

now, put p(x) = 0

so, 2x - 1 = 0
2x = 1
x = 1/2 ( alpha - a )
and,
3x - 2 = 0
3x = 2
x = 2/3 ( beta - b )

now,
1/a ( a* ) = 1/1/2
= 2
and,
1/b ( b* )= 1/2/3
= 3/2

now,

a* + b* = 2 + 3/2
= 7/2
and,
a* × b* = 2 × 7/2
= 7

so,
the quad. polynomial
= k [ x^2 - ( a* + b* ) + ( a*× b*) ]
= k [ x^2 - 7/2x + 7 ]