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. 2x⁵+7x⁴−18x²−8x+8=0

Verified step by step guidance

1. Identify the polynomial equation: $\mathrm{2x^5\; +\; 7x^4\; -\; 18x^2\; -\; 8x\; +\; 8\; =\; 0}$. Our goal is to find all zeros (roots) of this polynomial.
2. Apply the Rational Zero Theorem to list all possible rational zeros. The theorem states that any rational zero, expressed as $\backslash frac\{p\}\{q\}$, has $p$ as a factor of the constant term (8) and $q$ as a factor of the leading coefficient (2). List all such possible values.
3. Use Descartes's Rule of Signs to determine the possible 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 to narrow down the possibilities.
4. Test the possible rational zeros from step 2 by substituting them into the polynomial or using synthetic division to check which values yield zero. Once a zero is found, factor it out from the polynomial to reduce its degree.
5. Repeat the process on the reduced polynomial: apply the Rational Zero Theorem and Descartes's Rule of Signs again if necessary, and continue factoring until all zeros are found or the polynomial is factored completely.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem helps identify possible rational roots of a polynomial by considering factors of the constant term and the leading coefficient. These candidates can 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 transformed version. This helps narrow down the possible number of roots to check.

Polynomial Graphing and Root Approximation

Graphing a polynomial function using a graphing utility visually reveals the approximate locations of zeros. This aids in identifying initial roots and understanding the behavior of the polynomial, which supports algebraic methods in finding exact solutions.