4. Polynomial Functions

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

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

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)=2x³−3x²−11x+6

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

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)=4x^3-37x^2+50x+60 Find the zero in part (b) to three decimal places.

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 $3$)

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 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=(x+1{)}^2(x-1{)}^3(x^2-10)$

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1

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

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

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

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