Use the boundedness theorem to determine whether the statement is true or false.
-The polynomial $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero less than $- 1$.

Question

Quizplus · Accepted Answer

The polynomial $f(x) = x^4 - x^3 + 2x^2 - 4x - 10$ is continuous and differentiable everywhere, and its leading term $x^4$ ensures that as $x$ approaches negative infinity, $f(x)$ approaches positive infinity. Therefore, there must be a real zero less than $-1$ because $f(-1) = -1 - (-1) + 2 - 4 - 10 = -12$, indicating a negative value at $x = -1$, and since the polynomial must cross the x-axis to reach positive values as $x$ approaches negative infinity, there must be at least one real zero less than $-1$.

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