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. First, write down the polynomial function: \(f(x) = x^{3} - 4x^{2} - 7x + 10\).
2. Use the Rational Zero Theorem to list all possible rational zeros. These are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (10) and \(q\) divides the leading coefficient (1). So possible zeros are \(\pm1, \pm2, \pm5, \pm10\).
3. Apply Descartes's Rule of Signs to determine 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 ones yield zero. This will help identify at least one root.
5. Once a root \(r\) is found, use polynomial division or synthetic division to divide \(f(x)\) by \((x - r)\), reducing the polynomial to a quadratic. Then solve the quadratic equation to find the remaining zeros.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem helps identify all possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. These possible zeros are tested to find actual roots, 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 the polynomial and its transformed version f(-x). This rule guides the search for roots by narrowing down the possible number of positive and negative solutions.

Polynomial Graphing and Root Approximation

Graphing a polynomial function using a graphing utility provides a visual representation of the function's behavior and approximate locations of zeros. This aids in identifying initial roots, which can then be refined algebraically or numerically for exact solutions.