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. f(x)=x⁴−2x³+x²+12x+8

Verified step by step guidance

1. Identify the polynomial function: $f(x)=x^4-2x^3+x^2+12x+8$.
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 $f(x)$ to find at least one zero (root). Once a zero is found, use polynomial division or synthetic division to divide $f(x)$ by the corresponding factor $(x\; -\; \backslash text\{zero\})$ to reduce the polynomial degree.
5. Repeat the process with the reduced polynomial to find the remaining zeros, using the Rational Zero Theorem and Descartes's Rule of Signs as needed, until all zeros are found.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem helps identify all possible rational roots of a polynomial 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 coefficients of f(x) and f(-x). 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 verifying the number and nature of solutions, complementing algebraic methods.