Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=4x³−8x²−3x+9

Verified step by step guidance

1. Start by identifying the polynomial function given: \(f(x) = 4x^{3} - 8x^{2} - 3x + 9\).
2. To find the zeros of the function, set \(f(x) = 0\), which gives the equation \$4x^{3} - 8x^{2} - 3x + 9 = 0$.
3. Use the Rational Root Theorem to list possible rational zeros. These are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (9) and \(q\) divides the leading coefficient (4). So possible roots are \(\pm1, \pm3, \pm9, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{9}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{9}{4}\).
4. Test these possible roots by substituting them into the polynomial to find which values make \(f(x) = 0\). Once a root is found, use polynomial division or synthetic division to factor the polynomial and reduce its degree.
5. After factoring completely, solve the remaining quadratic or linear factors to find all zeros. Then, use the zeros and the end behavior of the cubic function to sketch the complete graph without a graphing utility.

Key Concepts

Finding Zeros of Polynomial Functions

Zeros of a polynomial are the values of x for which the function equals zero. To find them, one can use factoring, synthetic division, or the Rational Root Theorem to identify possible roots. These zeros correspond to the x-intercepts on the graph.

Behavior of Polynomial Graphs

Understanding the shape of a polynomial graph involves analyzing its degree and leading coefficient. The degree determines the number of turning points and end behavior, while the leading coefficient affects whether the graph rises or falls at the extremes.

Sketching Polynomial Graphs Without Technology

To sketch a polynomial graph by hand, identify zeros, determine their multiplicities, analyze end behavior, and find critical points using derivatives if possible. This approach helps create an accurate representation without relying on graphing utilities.