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

In summary, it is not possible for a cubic polynomial to have three real roots with one of them being infinity. This is because infinity is not an element of the real number domain. Even if we consider a limit as x approaches infinity, the polynomial will either go to positive or negative infinity, making it impossible for it to have a root at infinity. This reasoning applies to any odd-degree polynomial and for even-degree polynomials, the limit at infinity will also be infinity.
  • #1
hiroman
7
0
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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  • #2
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.
 
  • #3
Many thanks for the enlightenment, Steve! Cheers.
 

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

What is a cubic polynomial function with 3 real roots?

A cubic polynomial function is a mathematical equation of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are coefficients and x is the variable. A cubic polynomial function with 3 real roots means that when the function is graphed, it will have 3 points where the graph intersects the x-axis, indicating 3 solutions to the equation.

How do you determine the roots of a cubic polynomial function?

To find the roots of a cubic polynomial function, you can use the rational root theorem, synthetic division, or the cubic formula. These methods involve finding factors or solving equations to determine the values of x that make the function equal to 0.

What does it mean for one root to be at infinity?

If one root of a cubic polynomial function is at infinity, it means that the graph of the function will never intersect the x-axis and continues to increase or decrease without bound. In other words, the function has no real root at that point.

Can a cubic polynomial function have more than 3 real roots?

No, a cubic polynomial function can only have a maximum of 3 real roots. This is because a cubic function has a degree of 3, meaning it can have a maximum of 3 solutions to the equation f(x) = 0.

How do the roots of a cubic polynomial function affect its graph?

The roots of a cubic polynomial function have a direct impact on the graph. When a root is positive, the graph will intersect the x-axis at that point. When a root is negative, the graph will touch but not cross the x-axis. The number of roots also determines the overall shape of the graph, with 3 real roots creating a "S" shape and 2 real roots creating a "U" shape.

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