If $\left(a+b+c\right)>0$ and $<0
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Real and distinct roots
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Roots are imaginary
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Product of roots is negative
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Products of roots is positive
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Correct answer
AI explanation
Expanding the given equation $a(x-b)(x-c) + b(x-c) + c(x-a)(x-b) = 0$ results in a quadratic equation where the leading coefficient is $(a+c)$ and the constant term evaluates to $abc$. The product of the roots of this quadratic equation is given by dividing the constant term by the leading coefficient, which results in $abc / (a+c)$. Because the problem specifies that $0 < b < c$ and $(a+b+c) > 0$ (though logically $a$ must be positive to maintain the inequalities with $b$ and $c$), all components in the fraction $abc / (a+c)$ are positive. Therefore, the product of the roots must be positive.