4. Polynomial Functions

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=5x^2(x^2-16)(x+5)

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. 2x^5+7x^4−18x^2−8x+8=0

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, 3, and -1

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x+6=0

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=5x^4+3x^2+2x-9

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=2x^5-x^4+x^3-x^2+x+5

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=-2x^5+10x^4-6x^3+8x^2-x+1

Find the average rate of change of f(x)=√x from x1=4 to x2=9.

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^5-6x^4+14x^3-20x^2+24x-16

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=2x^4-x^3+7x^2-4x-4

Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)^4(x-3), the number 2 is a zero of multiplicity 4.

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x^4−x^3+5x^2−2x−6

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3+6x^2-2x-7; x+1

If ƒ(x) is a polynomial function with real coefficients, and if 7+2i is a zero of the function, then what other complex number must also be a zero?

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; -5 and 4+3i are zeros; f(2) = 91