Final answer:
The statement is false; while even degree polynomials can be even functions, they are not always so. An even function must satisfy f(x) = f(-x) for all x, which not all even degree polynomials do.
Step-by-step explanation:
The statement that an even degree polynomial is always an even function is false. An even function is defined as a function that satisfies f(x) = f(-x) for all x in the domain. The term even degree polynomial refers to the highest power of x within the polynomial being an even number. While it is true that the quadratic function, which is a second-order polynomial, is an even degree polynomial and can be an even function (e.g., f(x) = x²), not all even degree polynomials are even functions.
For example, the polynomial f(x) = x´ + x² + 1 is of even degree (4), but it is not an even function because it doesn't satisfy the even function property f(x) = f(-x) for all x. This is clear when x=1 and x=-1 are plugged into the polynomial: f(1) = 1 + 1 + 1 = 3 and f(-1) = 1 + 1 + 1 = 3, showing that it is an even function, but if we had a term like x or x³ added to the polynomial, it would break its evenness. Therefore, it's important to check the full expression of a polynomial to determine if it's an even function.