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)=x³+2x²+x-10

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x⁵-x³+7x-5

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+2x³-3x²+24x-180

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+4x³+6x²+4x+1

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+2x²+1

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-6x³+7x²

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-8x³+29x²-66x+72

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁶-9x⁴-16x²+144

Determine whether each statement is true or false. If false, explain why. The polynomial function $\text{ƒ}\left(x\right)=2x^5+3x^4-8x^3-5x+6$ has three variations in sign.

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x⁵−x⁴−7x³+7x²−12x−12

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3+4x^2−3x−6

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x³−2x²−11x+12=0

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

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