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)=3x^5+2x^4−15x^3−10x^2+12x+8

Verified step by step guidance

1. Step 1: Identify the degree of the polynomial. The given polynomial is \( f(x) = 3x^5 + 2x^4 - 15x^3 - 10x^2 + 12x + 8 \). The degree is 5, indicating there are 5 roots (including complex and repeated roots).
2. Step 2: Use the Rational Root Theorem to list possible rational roots. The possible rational roots are the factors of the constant term (8) divided by the factors of the leading coefficient (3).
3. Step 3: Test the possible rational roots using synthetic division or substitution to find actual roots. Start with simple values like \( x = \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3} \).
4. Step 4: Once a root is found, use synthetic division to divide the polynomial by \( x - \text{root} \) to reduce the polynomial's degree. Repeat the process to find all roots.
5. Step 5: After finding all zeros, use the factored form of the polynomial to sketch the graph. Consider the end behavior (since it's a degree 5 polynomial with a positive leading coefficient, it will start from the bottom left and end at the top right) and the multiplicity of roots to determine the shape of the graph at each zero.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer and 'a_n' are constants. Understanding the structure of polynomial functions is essential for identifying their zeros and behavior.

Finding Zeros

The zeros of a polynomial function are the values of 'x' for which f(x) = 0. These can be found using various methods, including factoring, synthetic division, or the Rational Root Theorem. Identifying the zeros is crucial for sketching the graph, as they indicate where the graph intersects the x-axis.

Graphing Techniques

Graphing a polynomial function involves plotting points based on the function's values and understanding its general shape. Key features to consider include the degree of the polynomial, the behavior at the ends (end behavior), and the multiplicity of the zeros, which affects how the graph behaves at those points. A complete graph provides a visual representation of the function's behavior across its domain.