Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=6x^4+13x^3-11x^2-3x+5no zero less than -3

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Step 1: Understand the problem. We need to show that the polynomial function \( f(x) = 6x^4 + 13x^3 - 11x^2 - 3x + 5 \) has a real zero, and that there is no zero less than -3.

Step 2: Use the Intermediate Value Theorem. This theorem states that if a continuous function changes sign over an interval \([a, b]\), then there is at least one real zero in that interval.

Step 3: Evaluate \( f(x) \) at specific points to find a sign change. Calculate \( f(-3) \) and \( f(-2) \) to check if there is a sign change between these points.

Step 4: Analyze the results. If \( f(-3) \) and \( f(-2) \) have opposite signs, then by the Intermediate Value Theorem, there is a real zero between \(-3\) and \(-2\).

Step 5: Conclude that there is no zero less than -3. If \( f(x) \) does not change sign for \( x < -3 \), then there is no zero in that region.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer and 'a_n' are constants. Understanding the behavior of polynomial functions, including their degree and leading coefficient, is essential for analyzing their roots or zeros.

Real Zeros

Real zeros of a polynomial function are the values of 'x' for which the function evaluates to zero, meaning f(x) = 0. These zeros can be found using various methods, including factoring, the Rational Root Theorem, or numerical methods. The existence of real zeros is crucial for understanding the function's graph and its intersections with the x-axis.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints (f(a) and f(b)), then there exists at least one c in (a, b) such that f(c) = 0. This theorem is instrumental in proving the existence of real zeros within specified intervals, particularly when analyzing polynomial functions.