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)=x³−4x²−7x+10

Verified step by step guidance

1. Identify the polynomial function: $f(x)=x^3-4x^2-7x+10$.
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 (10) and $q$ divides the leading coefficient (1). So possible zeros are $\backslash pm\; 1,\; \backslash pm\; 2,\; \backslash pm\; 5,\; \backslash pm\; 10$.
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 which values make the polynomial equal to zero. This will help identify at least one root.
5. Once a root $r$ is found, perform polynomial division (either synthetic or long division) to divide $f(x)$ by $(x\; -\; r)$. This will reduce the cubic polynomial to a quadratic, which can then be solved using factoring or the quadratic formula to find the remaining zeros.

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 predicts the number of positive and negative real zeros of a polynomial by counting sign changes in f(x) and f(-x). This rule guides the search for roots by narrowing down how many positive or negative solutions to expect.

Polynomial Graphing and Root Approximation

Graphing a polynomial function provides a visual representation of its behavior and approximate locations of zeros. Using a graphing utility helps identify initial roots, which can then be refined algebraically, aiding in solving higher-degree polynomials.