Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ 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, can help determine the range within which all real zeros of a polynomial must lie. However, it does not provide a definitive answer about the existence of zeros beyond a certain point without further analysis. The assertion that the polynomial $f(x) = x^5 + 10x^4 - 25x^3 - 10x^2 + 24x$ has no real zero greater than 3 cannot be directly concluded from the boundedness theorem alone. My explanation was incorrect because the boundedness theorem does not specifically determine the absence of zeros beyond a certain point without additional investigation or application of other theorems.

Use the Boundedness Theorem to Determine Whether the Polynomial Function f(x)=x5+10x4−25x3−10x2+24xf ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 xf(x)=x5+10x4−25x3−10x2+24x

Use the Boundedness Theorem to Determine Whether the Polynomial Function $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$