4. Polynomial Functions

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)=5x³-9x²+28x+6

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=4x³+3x²+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)=3x⁴−11x³−3x²−6x+8

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. 4x²+2x+54; x-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

Factor ƒ(x) into linear factors given that k is a zero. $\text{ƒ}\left(x\right)=2x^3-3x^2-5x+6{;}^{}k=1$

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

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x³+2x²+5x+4

For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=x^3+4x^2-5;1$

Solve each problem. Is x+1 a factor of ƒ(x)=x³+2x²+3x+2?

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $f\left(x\right)=3x^4-2x^3-8x+5$

Solve each problem. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ(2)=16.