4. Polynomial Functions

Factor $\text{ƒ}\left(x\right)$ into linear factors given that k is a zero. $\text{ƒ}\left(x\right)=2x^4+x^3-9x^2-13x-5;k=-1$ (multiplicity $3$)

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. x³−10x−12=0

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=6x^4+2x^3+9x^2+x+5$

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+5 no zero less than -3

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x−1=0

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁵-6x⁴+14x³-20x²+24x-16

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

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

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x³+x²−4x−4

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=2x^4-x^3+7x^2-4x-4$

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.

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.

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. x³+2x²+3; x-1

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

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

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)=3x^3-8x^2+x+2 between 2 and 3