Cubic polynomial function with 3 real roots; one at infinity?

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SUMMARY

A cubic polynomial of the form ax^3 + bx^2 + cx + d cannot have three real roots with one of them being at infinity. The reasoning is based on the behavior of cubic functions as x approaches positive or negative infinity; specifically, as x approaches +∞, the polynomial approaches +∞, and as x approaches -∞, it approaches -∞. Therefore, it is impossible for the limit of a cubic polynomial to equal zero at infinity.

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hiroman
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Is it possible to have a cubic polynomial (ax^3+bx^2+cx+d) which has three REAL roots, with one of them being +/- infinity?

If there is, can you give an example?

Thanks!
 
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hiroman said:
Is it possible to have a cubic polynomial (ax^3+bx^2+cx+d) which has three REAL roots, with one of them being +/- infinity?

If there is, can you give an example?

Thanks!

It doesn't make literal sense for any function whose domain is the real numbers to have a root at infinity, since infinity's not an element of the domain.

But we could still ask if there's a 3rd degree polynomial such that the limit as x->inf is 0. The answer's no, and it's easy to see. As x -> +inf, x^3 goes to +inf. Since the x^3 term eventually dominates the rest of the terms, the polynomial goes to +inf.

Likewise, as x -> -inf, the function must go to -inf. So it's not possible for the polynomial to go to zero at +/- infinity. This reasoning goes through for any odd-degree polynomial. For even-degree polynomials, the limits at +/- infinity are both infinity.
 
Many thanks for the enlightenment, Steve! Cheers.
 

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