Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x⁵-3x³+x+2; no real zero greater than 2

Verified step by step guidance

1. First, understand the problem: we need to show that the real zeros of the polynomial function \(f(x) = x^5 - 3x^3 + x + 2\) have no real zero greater than 2. This means if \(r\) is a real root, then \(r \leq 2\).
2. Evaluate the polynomial at \(x = 2\) to check the sign of \(f(2)\). Substitute \(x = 2\) into the polynomial: \(f(2) = 2^5 - 3 \cdot 2^3 + 2 + 2\).
3. Analyze the behavior of \(f(x)\) for values greater than 2. Consider the leading term \(x^5\), which dominates for large \(x\). Since the leading coefficient is positive, \(f(x)\) tends to \(+\infty\) as \(x \to +\infty\).
4. Use the Intermediate Value Theorem: if \(f(2)\) is positive and \(f(x)\) tends to \(+\infty\) for \(x > 2\), then there is no root greater than 2 because the function does not cross the x-axis beyond 2.
5. Optionally, check \(f(3)\) or another value greater than 2 to confirm the function remains positive, reinforcing that no real zeros exist greater than 2.

Key Concepts

Real Zeros of Polynomial Functions

Real zeros of a polynomial are the values of x for which the polynomial equals zero. These zeros correspond to the x-intercepts of the graph. Understanding how to find and interpret real zeros is essential for analyzing the behavior of polynomial functions.

Evaluating Polynomial Values to Bound Zeros

To show that no real zero exceeds a certain value, evaluate the polynomial at that value and analyze the sign of the result. If the polynomial is positive (or negative) at that point and the function’s behavior indicates no sign changes beyond it, this helps establish bounds on the zeros.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, it must have a zero within that interval. This theorem is useful for locating zeros and proving that no zeros exist beyond a certain point by checking sign consistency.