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)=2x⁴−3x³−7x²−8x+6

Verified step by step guidance

1. Start by finding the zeros of the polynomial function $f\left(x\right)=2x^4-3x^3-7x^2-8x+6$ by setting $f\left(x\right)=\; 0$ and solving for $x$.
2. Use the Rational Root Theorem to list possible rational zeros. These are of the form $\backslash pm\; \backslash frac\{p\}\{q\}$, where $p$ divides the constant term 6 and $q$ divides the leading coefficient 2. So possible roots are $\backslash pm\; 1,\; \backslash pm\; 2,\; \backslash pm\; 3,\; \backslash pm\; 6,\; \backslash pm\; \backslash frac\{1\}\{2\},\; \backslash pm\; \backslash frac\{3\}\{2\}$.
3. Test each possible rational root by substituting into the polynomial or using synthetic division to check if it yields zero. When a root is found, factor it out from the polynomial.
4. After factoring out the first root, reduce the polynomial degree and repeat the process to find other zeros. Continue factoring until the polynomial is completely factored into linear and/or irreducible quadratic factors.
5. Once all zeros are found, analyze the multiplicity of each zero to understand the behavior of the graph at those points. Use this information along with the leading coefficient and degree 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 test possible roots. Identifying all zeros is essential for understanding the behavior and shape of the 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, a positive leading coefficient with an even degree means the graph rises on both ends, guiding the sketching of the complete 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.