Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero greater than 3 .
A) Yes, the boundedness theorem shows that the polynomial has no real zero greater than 3 .
B) No, the boundedness theorem does not show that the polynomial has no real zero greater than 3.

Question

Quizplus · Accepted Answer

The boundedness theorem, also known as the polynomial bounds theorem, helps in determining the bounds within which all real zeros of a polynomial must lie. For the polynomial $f(x) = x^4 - x^3 + 2x^2 - 4x - 10$, as $x$ becomes very large, the $x^4$ term dominates, making the function positive for large positive $x$. This means that beyond a certain point, the function does not cross the x-axis, implying it has no real zeros beyond that point. Since the question specifically asks about real zeros greater than 3, and given the nature of the polynomial, it's reasonable to conclude based on the theorem that the polynomial does not have any real zeros greater than 3, without directly computing the zeros.

Use the Boundedness Theorem to Determine Whether the Polynomial Function f(x)=x4−x3+2x2−4x−10f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10f(x)=x4−x3+2x2−4x−10

Use the Boundedness Theorem to Determine Whether the Polynomial Function $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$