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. x⁴−3x³−20x²−24x−8=0

Verified step by step guidance

1. Identify the polynomial equation: $x{4}^{}-\; 3x^3-\; 20x^2-\; 24x-\; 8\; =\; 0$.
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 (8) and $q$ divides the leading coefficient (1). So possible rational zeros are $\backslash pm\; 1,\; \backslash pm\; 2,\; \backslash pm\; 4,\; \backslash pm\; 8$.
3. Use Descartes's Rule of Signs to estimate the number of positive and negative real zeros. For positive zeros, count sign changes in $f(x)$. For negative zeros, count sign changes in $f(-x)$.
4. Test the possible rational zeros by substituting them into the polynomial or using synthetic division to find a zero that makes the polynomial equal to zero.
5. Once a zero is found, use polynomial division (synthetic or long division) to divide the original polynomial by the corresponding factor $(x\; -\; \backslash text\{zero\})$ to reduce the polynomial degree and then solve the resulting polynomial equation for the remaining 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 candidates can then be tested to find actual zeros, 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 substitution with -x. This rule guides the search for roots and narrows down possible zero counts.

Polynomial Graphing and Root Approximation

Graphing a polynomial function using a graphing utility visually reveals the approximate locations of zeros, including multiplicities and intervals where roots lie. This aids in selecting initial guesses for root-finding methods and verifying the nature of the roots.