4. Polynomial Functions

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=3; -5 and 4+3i are zeros; f(2) = 91

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zero of -3 having multiplicity 3; ƒ(3)=36

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

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

For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. $f\left(x\right)=x^3+3x^2-4$

For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=3x(x-2)(x+3)(x^2-1)$

For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=4x^3+6x^2-2x-1;\frac{1}{2}$

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x⁴−3x²−4). If f(3)=−150, determine the value of a_n.

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=2x^5+11x^4+16x^3+15x^2+36x$

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x³-9x²+28x+6

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

$x^3+6x^2-2x-7;x+1$

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

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

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

For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=x^3-x^2-4x-6;3$