Use the boundedness theorem to determine whether the statement is true or false.
-The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero less than $- 8$.

Question

Quizplus · Accepted Answer

The polynomial $f(x) = x^5 + 10x^4 - 25x^3 - 10x^2 + 24x$ is not bounded by the conditions of the boundedness theorem, which typically applies to sequences or functions within a certain interval. To determine the real zeros of a polynomial, one would typically use methods such as factoring, graphing, or applying the Rational Root Theorem, not the boundedness theorem. Moreover, without specific analysis or graphing, we cannot assert the absence of real zeros below a certain value just based on the polynomial's expression.

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