In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 4x⁴−x³+5x²−2x−6=0

Verified step by step guidance

1. First, identify the polynomial equation: $\mathrm{4x^4}-\mathrm{x^3}+\mathrm{5x^2}-\mathrm{2x}-6=\; 0$.
2. Apply the Rational Zero Theorem to list all possible rational zeros. These are of the form $\backslash pm\; \backslash frac\{p\}\{q\}$, where $p$ divides the constant term (6) and $q$ divides the leading coefficient (4). So, possible values for $p$ are ±1, ±2, ±3, ±6, and for $q$ are 1, 2, 4. Combine these to get all possible rational zeros.
3. Use Descartes's Rule of Signs to determine the number of positive and negative real zeros. Count the number of sign changes in $f(x)$ for positive zeros, and in $f(-x)$ for negative zeros. This helps narrow down which candidates from the Rational Zero Theorem to test first.
4. Test the possible rational zeros by substituting them into the polynomial or using synthetic division to check if they yield zero. Once a zero is found, factor it out of the polynomial to reduce the degree.
5. Repeat the process with the reduced polynomial to find the remaining zeros. If necessary, use the quadratic formula or factoring techniques to solve the lower-degree polynomial obtained after factoring out the known zeros.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem helps identify all possible rational roots of a polynomial equation by considering factors of the constant term and the leading coefficient. These possible zeros are tested to find actual roots, simplifying the process of solving higher-degree polynomials.

Descartes's Rule of Signs

Descartes's Rule of Signs provides a way to estimate the number of positive and negative real zeros of a polynomial by counting sign changes in the polynomial and its transformed version. This rule guides the search for roots and narrows down the possibilities.

Polynomial Graphing and Root Approximation

Graphing a polynomial function using a graphing utility visually reveals the approximate locations of its zeros. This aids in identifying initial roots, which can then be refined algebraically, making it easier to solve complex polynomial equations.