Final answer:
The question asks about confirming if a given mathematical expression is even, given that xy is even. The expression involves exponents and the properties of even and odd functions. When simplifying the given expression, assuming no typos and clarifying ambiguities, the result of the expression will indeed be even if xy is even.
Step-by-step explanation:
The question addresses a mathematical statement regarding the properties of even and odd functions in relation to polynomials. We know that an even function times an even function results in an even function, and an odd function times an odd function also produces an even function. Conversely, the multiplication of an odd function with an even function yields an odd function. In the given expression xy² - x³ 4, provided that xy is even, we want to determine if the resulting expression is even. To solve this, we would look at the properties of exponents and parity.
The misconception might arise from misinterpretation or a typographical error in the expression. If it is interpreted as (xy)² - (x³) (4), we recognize that the square of any number, even or odd, will be even. As such, (xy)² is even. Since x³ is raised to an odd power, it retains the parity of x. However, when x³ is multiplied by 4, an even number, the result is even regardless of whether x³ was odd or even. Therefore, the expression consists of the subtraction of two even numbers, which will always result in an even number.