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Consider the polynomial

(x^2 - 47x + 1)(x^2 - 47x + 2)(x^2 - 47x + 3) ... (x^2 - 47x + 999)

 

What is the product of the real roots of this polynomial?  I don't know what to do here.

 Dec 30, 2020
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First, notice that the quadratics in the expression only have real roots when its discriminant is nonnegative. The biggest possible value for the constant term is 552 because when it is 553, there will be no real roots.

 

The product of the roots of a quadratic is equal to the constant term divided by the quadratic term. Since the quadratic term is always 1, the product of the roots of these quadratics will always be equal to the constant term.

 

Therefore, the problem is reduced to the product of 1*2*3*4*5*6*....*551*552.

That is equal to 552! (552 factorial).

 

That number is giant, so IDK if you want me to type it out.

 Dec 31, 2020

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