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. Identify the polynomial function given: $f(x)=3x^5\; +\; 2x^4\; -\; 15x^3\; -\; 10x^2\; +\; 12x\; +\; 8$.
2. To find the zeros of the function, start by looking for rational roots using the Rational Root Theorem. List possible rational roots as factors of the constant term (8) over factors of the leading coefficient (3). These possible roots are ±1, ±2, ±4, ±8, ±1/3, ±2/3, ±4/3, ±8/3.
3. Test each possible rational root by substituting into the polynomial or by using synthetic division to check if it yields zero. When a root is found, factor it out from the polynomial to reduce the degree.
4. Repeat the process of finding roots and factoring until the polynomial is completely factored into linear and/or irreducible quadratic factors. This will give all zeros of the function, including their multiplicities.
5. Use the zeros and their multiplicities to sketch the graph: plot the zeros on the x-axis, determine the end behavior by considering the leading term $3x^5$, and analyze the behavior near each zero (crossing or touching the x-axis) to complete the 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.

Polynomial Function Graphing

Graphing a polynomial involves plotting its zeros, determining end behavior based on the leading term, and analyzing turning points. Understanding how the degree and leading coefficient affect the graph helps in sketching an accurate curve without technology.

End Behavior of Polynomials

The end behavior describes how the function behaves as x approaches positive or negative infinity. It depends on the degree and leading coefficient: for odd-degree polynomials, ends go in opposite directions; for even-degree, both ends go the same way. This guides the overall shape of the graph.