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)=3x⁵+2x⁴−15x³−10x²+12x+8

Verified step by step guidance

1. Start by finding the zeros of the polynomial function \(f(x) = 3x^{5} + 2x^{4} - 15x^{3} - 10x^{2} + 12x + 8\). To do this, first look for possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term 8 and \(q\) divides the leading coefficient 3.
2. Test the possible rational zeros by substituting them into the polynomial or by using synthetic division to check if they yield a remainder of zero. Each zero found will correspond to a factor of the polynomial.
3. Once a zero is found, use polynomial division (either long division or synthetic division) to divide the original polynomial by the corresponding factor \((x - r)\), where \(r\) is the zero. This will reduce the polynomial to a lower degree, making it easier to find the remaining zeros.
4. Repeat the process of finding zeros and dividing the polynomial until you factor the polynomial completely into linear and/or irreducible quadratic factors. The zeros of the polynomial are the roots of these factors.
5. After finding all zeros, analyze the multiplicity of each zero to understand the behavior of the graph at those points (whether the graph crosses or touches the x-axis). Then, use this information along with the end behavior of the polynomial (determined by the leading term \$3x^{5}$) to sketch a complete graph of the function without using 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 typically factors the polynomial or uses methods like synthetic division or the Rational Root Theorem. Identifying all zeros is essential for understanding the function's behavior and graph.

End Behavior of Polynomial Functions

The end behavior describes how the function behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient. For example, an odd-degree polynomial with a positive leading coefficient rises to the right and falls to the left, guiding the sketch of the graph.

Sketching Polynomial Graphs Without Technology

Drawing a polynomial graph by hand involves plotting zeros, determining multiplicities, analyzing end behavior, and finding key points like local maxima and minima. Understanding these features helps create an accurate, complete graph without relying on graphing utilities.