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)=3x⁴−11x³−x²+19x+6

Verified step by step guidance

1. First, write down the polynomial function: \(f(x) = 3x^{4} - 11x^{3} - x^{2} + 19x + 6\).
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 (6) and \(q\) divides the leading coefficient (3). So possible values for \(p\) are \(\pm1, \pm2, \pm3, \pm6\) and for \(q\) are \(\pm1, \pm3\).
3. Form the list of possible rational zeros: \(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{3}, \pm\frac{2}{3}\).
4. Apply 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 \(f(-x)\).
5. Test the possible rational zeros by substituting them into \(f(x)\) or use synthetic division to find a zero. Once a zero is found, use polynomial division to factor it out and reduce the polynomial to a cubic or quadratic, then repeat the process to find all zeros.

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 polynomial and its transformed form. This rule narrows down the possible number of real roots, guiding the search for zeros.

Graphing Polynomial Functions

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