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If euation x^(3) + ax^(2) + bx + c = 0, ...

If euation `x^(3) + ax^(2) + bx + c = 0,` where a, b, c `in Q (a ne `1). If the real roots of the equation are `x_(1), x_(2) and x_(1)x_(2),` then prove that `x_(1)x_(2)` is rational.

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We have
`x_(1) +x_(2)+ x_(1)x_(2) = -a`
`x_(1) +x_(2)+ x_(1)x_(2)^(2)+x_(2)x_(1)^(2) = b`
and ` (x_(1)x_(2))^(2) = -c`
`therefore x_(1)x_(2)(1+ x_(1)+x_(2))=b`
`rArr x_(1)x_(2)(1+ a+x_(1)x_(2))=b`
`rArr (1 -a)x_(1)x_(2)-(x_(1)x_(2))^(2)=b`
`rArr (1 -a)x_(1)x_(2) + c = b`
`rArr x_(1)x_(2) =(b- c )/(1- b),` which is clearly a rational number.
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