4. Polynomial Functions

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.

$5x^4+16x^3-15x^2+8x+16;x+4$

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x³−3x²+3x−1

Use Descartes' Rule of Signs to explain why $2x^4+6x^2+8=0$ has no real roots.

Solve each problem. Give the maximum number of turning points of the graph of each function. ƒ(x)=4x^3-6x^2+2

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x⁴−3x³−20x²−24x−8=0

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

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)=−x³+x²+16x−16

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

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x⁴−3x²−4). If f(3)=−150, determine the value of a_n.