4. Polynomial Functions

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $f\left(x\right)=x^4-6x^3+14x^2-14x+5$

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; 1 and 5i are zeros; f(-1) = -104

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

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. 2x³−x²−9x−4=0

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=7x^5+6x^4+2x^3+9x^2+x+5$

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

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x⁴−5x³−x²−6x+4

Solve: $x^4-6x^3+4x^2+15x+4=0$.

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=5x^6-6x^5+7x^3-4x^2+x+2$

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. x³-5x²+3x+1; x-1

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=x^4+x^3-9x^2+11x-4$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x⁴−11x³−x²+19x+6

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 greater than 1

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)=x³+x²−4x−4

Use Descartes' Rule of Signs to explain why $2x^4+6x^2+8=0$ has no real roots.