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+5 no zero greater than 1

Verified step by step guidance

1. insert step 1: Use the Intermediate Value Theorem to determine if there is a real zero between two values.
2. insert step 2: Evaluate the function \( f(x) = 6x^4 + 13x^3 - 11x^2 - 3x + 5 \) at \( x = 1 \) and \( x = 0 \).
3. insert step 3: Check the sign of \( f(x) \) at these points to see if there is a sign change, indicating a zero between them.
4. insert step 4: Use synthetic division or the Rational Root Theorem to test possible rational roots between 0 and 1.
5. insert step 5: Confirm that no zero is greater than 1 by evaluating \( f(x) \) at values greater than 1 and checking for sign changes.

Key Concepts

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, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is fundamental in proving the existence of real zeros for polynomial functions, especially when combined with the knowledge of their behavior at specific points.