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. 2x^3+x+2; x+1

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

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=-6x^3-25x^2-3x+4; k=-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)=5x^3−3x^2+3x−1

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=4x^3+6x^2-2x-1; 1/2

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. x^4−3x^3−20x^2−24x−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.

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x^2+x-2)^5(x-1+√3)^2

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)=−x^3+x^2+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)=2x^4−3x^3−7x^2−8x+6

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 2-i and 6-3i

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=2x^3-4x^2+2x+7

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x^4−3x^2−4). If f(3)=−150, determine the value of a_n.