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)=2x^4−3x^3−7x^2−8x+6

Verified step by step guidance

1. Step 1: Identify the degree of the polynomial. The given polynomial is $f(x) = 2x^4 - 3x^3 - 7x^2 - 8x + 6$. The degree is 4, which means there are 4 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 (6) divided by the factors of the leading coefficient (2).
3. Step 3: Test the possible rational roots using synthetic division or direct substitution to find actual roots. Start with simple values like $x = 1, -1, 2, -2, 3, -3, 6, -6, \frac{1}{2}, -\frac{1}{2}, \frac{3}{2}, -\frac{3}{2}$.
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 other roots.
5. Step 5: After finding all roots, use them to sketch the graph. Consider the end behavior of the polynomial (since it's degree 4 and the leading coefficient is positive, the graph will rise to the right and left). Plot the roots on the x-axis and sketch the curve, considering multiplicity of roots and turning points.

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 crucial 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, the Rational Root Theorem, or synthetic division. Identifying the zeros is essential 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 leading coefficient, and the behavior at the zeros and endpoints. Techniques such as identifying intercepts, end behavior, and turning points help create an accurate representation of the function.