Chapter 4 Review 1) 2) 3) Use the rational root thm to list all possible rational roots for the function below. f (x) = 6x 4 − x + 15 Find a 3rd degree polynomial with integer coefficients whose roots include –5 and 3 – 2i. Answer each of the following as True (T) or False (F): An even-­‐degreed polynomial with no real roots has no x-­‐intercepts. Any odd-­‐degreed polynomial must have at least one x-­‐intercept. All polynomial functions have exactly one y-­‐intercept. If a polynomial of degree n only has n-­‐1 zeros, then it must have a double root. The axis of symmetry for a standard parabolic function is given by y = k. 4) List the specific ways in which the numbers, types, and multiplicities of roots for a fifth degree equation could combine to form a polynomial that has exactly two x-­‐intercepts. 5) For the parabola given at right, complete the following: y = −3x 2 +12x −13 eq’n in std form: Coordinates of the Vertex = Circle one relative to the vertex: max / min Eq’n for the axis of Symmetry à Accurately graph the parabola at right à 4
2
–5
5
–2
–4
6) Sketch the graph of a generic fifth degree polynomial with a double root at 3 and a triple root at –1. Assume a negative leading coefficient. 8) Perform the following calculation: 7) Without performing the actual division, use the remainder theorem to find the remainder of the given quotient. 3
x 4 − 3x 3 + 2x −12
. x−5
x − 2x − 8
x+3
9) Find all zeroes of the function f(x) = x4 + x3 – 5x2 – 3x + 6 given that x – 3 is a factor. 10) Factor f (x) = 2x 4 + 5x 3 − x 2 + 5x − 3 completely over the complex numbers. (That is, write it as a product of linear factors.) 11) Find the values of a and b such that x + 3 and x – 2 will be exact factors of the polynomial given by f(x) = 2x4 – ax2 – bx – 24.