Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
$\text{ƒ}\left(x\right)=3x^4+2x^3-4x^2+x-1$; no real zero less than -2

Verified step by step guidance

1. First, understand the problem: we need to show that the real zeros of the polynomial function \(f(x) = 3x^4 + 2x^3 - 4x^2 + x - 1\) are all greater than or equal to \(-2\), meaning there are no real zeros less than \(-2\).
2. Evaluate the polynomial at \(x = -2\) to check the sign of \(f(-2)\). Substitute \(x = -2\) into the polynomial: \(f(-2) = 3(-2)^4 + 2(-2)^3 - 4(-2)^2 + (-2) - 1\).
3. Next, analyze the behavior of \(f(x)\) for values less than \(-2\). One way is to check the sign of \(f(x)\) at a value less than \(-2\), for example at \(x = -3\), by substituting \(x = -3\) into the polynomial.
4. If \(f(-2)\) and \(f(-3)\) have the same sign (both positive or both negative), then by the Intermediate Value Theorem, there is no zero between \(-3\) and \(-2\). Repeat this for other values less than \(-2\) if necessary to confirm no sign changes occur.
5. Finally, conclude that since the polynomial does not change sign for \(x < -2\), there are no real zeros less than \(-2\). This shows all real zeros satisfy the condition of being greater than or equal to \(-2\).

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. Finding real zeros often involves factoring, using the Rational Root Theorem, or applying numerical methods.

Evaluating Polynomial Values to Test Inequalities

To verify conditions on zeros, such as no real zero less than a certain value, evaluate the polynomial at strategic points and analyze sign changes. This helps determine intervals where zeros may lie, using the Intermediate Value Theorem.

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 essential for locating zeros and confirming their position relative to given bounds.