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. 2x³−x²−9x−4=0

Verified step by step guidance

1. Identify the polynomial equation: $\mathrm{2x^3}-\mathrm{x^2}-\mathrm{9x}-4=0$.
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 (4) and $q$ divides the leading coefficient (2). So possible values for $p$ are ±1, ±2, ±4 and for $q$ are ±1, ±2, giving possible rational zeros: ±1, ±2, ±4, ±1/2.
3. Use Descartes's Rule of Signs to estimate the number of positive and negative real zeros. For positive zeros, count sign changes in $2x^3\; -\; x^2\; -\; 9x\; -\; 4$. For negative zeros, substitute $x\; =\; -x$ and count sign changes in $2(-x)^3\; -\; (-x)^2\; -\; 9(-x)\; -\; 4$.
4. Test the possible rational zeros from step 2 by substituting them into the polynomial to find which ones yield zero. This can be done by direct substitution or synthetic division.
5. Once a zero is found, use polynomial division (synthetic or long division) to divide the original polynomial by the corresponding factor $(x\; -\; r)$, where $r$ is the zero found, to reduce the polynomial to a quadratic. Then solve the quadratic by factoring, completing the square, or using the quadratic formula to find 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 polynomial equations.

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 negative variable. This helps narrow down the possible number of roots to check.

Polynomial Root Finding and Graphing Utilities

Graphing utilities visually display the polynomial function, allowing identification of approximate roots and behavior of the graph. This visual aid supports analytical methods by suggesting initial guesses for zeros and confirming the number and nature of roots.