Final answer:
The polynomial F(x) = x⁴ - x³ - 2x² is a fourth-degree polynomial with a graph trending to positive infinity as x goes to infinity and has up to three turning points. The polynomial F(x) = (x-1)²(x+1) is a third-degree polynomial with x-intercepts at x = 1 and x = -1 and ends going off in opposite directions due to its odd degree.
Step-by-step explanation:
Properties of Polynomial Graphs
The properties of the graph of polynomial functions can be complex, varying with the degree of the polynomial and the coefficients of the terms. Let's examine the given functions separately:
A.) For the polynomial function F(x) = x⁴ - x³ - 2x², it is already in standard form. This is a fourth-degree polynomial, and its graph will exhibit properties characteristic of even-degree polynomials. Specifically, as x approaches positive and negative infinity, F(x) will tend to positive infinity due to the dominant x⁴ term. The function has a possibility of having multiple turning points, which can be a maximum of three (one less than the degree of the polynomial). Looking for the x-intercepts is more complex but can be done by setting F(x) to zero and solving for x.
B.) For the polynomial function F(x) = (x-1)²(x+1), this is a third-degree polynomial when expanded but given in factored form, which aids in finding the x-intercepts quickly, at x = 1 and x = -1. The squared term indicates a multiplicity, which means the graph will touch the x-axis at x = 1 but will not cross it. Since it's an odd-degree polynomial, the ends of the graph will go off in opposite directions; as x approaches infinity, F(x) will tend to infinity, and as x approaches negative infinity, F(x) will tend to negative infinity.
Understanding these properties helps us sketch the rough shape of polynomial graphs even before using an equation grapher to see the precise curve. The constants in the equation impact the stretch or compression of the graph but do not affect the fundamental shape dictated by the polynomial's degree and the sign of the leading coefficient.