4. Polynomial Functions

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=6x^4+13x^3-11x^2-3x+5no zero less than -3

Solve each problem. Use Descartes' rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of ƒ(x)=x^3+3x^2-4x-2.

In Exercises 35–36, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x) = x^4 - 6x^3 + 14x^2 -14x + 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. f(x)=3x^4−11x^3−x^2+19x+6

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. f(x) = 2x^4 + 3x^3 + 3x - 2

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

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x−1=0

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

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

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=9x^6-7x^4+8x^2+x+6

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=7x^5+6x^4+2x^3+9x^2+x+5

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=2x^5+11x^4+16x^3+15x^2+36x

Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x^6-x^4+2x^2-2, we can also conclude that ƒ(1)=0

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−x^2+19x+6

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3-5x^2+3x+1; x-1