Multiply out $a(x-x_1)(x-x_2)(x-x_3)$ and note the constant term... – 2011-11-28
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$p(x)=a(x-x_1)(x-x_2)(x-x_3)$ – 2011-11-28
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I think there's a negative sign missing... – 2011-11-28
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Just so everything's above the board: the image that originally accompanied the question had a minus sign missing. Check the edit history. – 2011-11-28
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It's not true without further hypotheses, e.g. mod $6$ note $x^3-x$ has roots $x = -1,1,2$ but their product $\rm -2 \not\equiv 0$. – 2011-11-28
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@BillDubuque um... The roots of $x^3-x$ are 0,1,-1 ... Or did you mean to write something else? – 2012-01-05
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@BillDubuque Oh, nevermind, I misunderstood. I think though that we can safely assume that the OP meant polynomials over a field. – 2012-01-05
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@Tim Even better, Bill's example is over a ring, not a field :-). In fact, the statement is true over all fields. – 2012-01-05