Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
$\text{ƒ}\left(x\right)=x^5-3x^3+x+2$; no real zero less than -3

Verified step by step guidance

1. First, understand the problem: we need to show that the polynomial function \(f(x) = x^5 - 3x^3 + x + 2\) has no real zeros less than \(-3\). This means that for all \(x < -3\), \(f(x) \neq 0\).
2. Step 1: Evaluate the function at \(x = -3\) to get a reference value. Calculate \(f(-3) = (-3)^5 - 3(-3)^3 + (-3) + 2\).
3. Step 2: Analyze the behavior of \(f(x)\) for values less than \(-3\). Since \(f(x)\) is a polynomial, it is continuous everywhere. If there were a zero less than \(-3\), the function would have to cross the x-axis somewhere in that region.
4. Step 3: Check the sign of \(f(x)\) at a point less than \(-3\), for example at \(x = -4\). Calculate \(f(-4) = (-4)^5 - 3(-4)^3 + (-4) + 2\) and note its sign.
5. Step 4: Use the Intermediate Value Theorem: if \(f(-4)\) and \(f(-3)\) have the same sign, then there is no zero between \(-4\) and \(-3\). Since the polynomial is continuous, and if this pattern holds for all \(x < -3\), it shows no zeros exist less than \(-3\).

Key Concepts

Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x for which the function equals zero. Understanding how to find and interpret these zeros is essential for analyzing the behavior of the function.

Evaluating Polynomial Values to Determine Zero Locations

To show that no real zero lies below a certain value, evaluate the polynomial at that value and nearby points. If the polynomial does not change sign below that point, it indicates no zeros exist there. This method uses the Intermediate Value Theorem and sign analysis to locate zeros.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, it must have at least one zero in that interval. This theorem helps confirm the existence or absence of zeros within specific ranges by checking function values at interval endpoints.