Final answer:
To find the value of 'a' when 'x - a' is a factor of the polynomial 'x⁵ - a²x³ + 2x + a + 3', we use the Remainder Theorem and solve for 'a' in the equation 'a⁵ - a²a³ + 2a + a + 3 = 0'.
Step-by-step explanation:
If x - a is a factor of the polynomial x⁵ - a²x³ + 2x + a + 3, we can use polynomial long division or synthetic division to find the value of a. However, a faster way to find a is by applying the Remainder Theorem, which states that if x - a is a factor of a polynomial, then the polynomial evaluated at x = a will equal zero. Thus, we replace every x in the polynomial with a and solve for a.
The equation becomes a⁵ - a²a³ + 2a + a + 3 = 0. Simplifying this yields a(a⁴ - a² + 3) = -3, and from this equation, we would find the values of a that satisfy the equation. Given that the expression needs to be equal to zero, one obvious potential solution for a is 0. However, since a is squared in the original polynomial, zero is not an acceptable value (because it would result in the term - a² x³ to be zero as well). Hence, we need to find another value for a that, when plugged into the polynomial, yields a zero remainder. This typically involves solving the simplified polynomial explicitly for a.