4. Polynomial Functions

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

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

Find all zeros of f(x) = x³ + 5x² – 8x + 2.

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x^3-9x^2+28x+6

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=4x^3+3x^2+8x+6

Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−3x^2−6x+8

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 4x^2+2x+54; x-4

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

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

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=4; -2, 5, and 3+2i are zeros; f(1) = -96

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)=x^3+2x^2+5x+4

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=x^3+4x^2-5; 1

Solve each problem. Is x+1 a factor of ƒ(x)=x^3+2x^2+3x+2?

In Exercises 37–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x) = 3x^4 - 2x^3 - 8x + 5