Answer :
To solve the problem of finding the remainder when [tex]x^{83} + x^{65} + x^{47} + x^{29} + x^{20} + x^{11} + x^5[/tex] is divided by [tex]x^5 - x^3[/tex], we can utilize polynomial division.
Let's break this down step-by-step:
Factor the divisor: Notice that [tex]x^5 - x^3[/tex] can be factored as [tex]x^3(x^2 - 1) = x^3(x - 1)(x + 1)[/tex].
Remainder Theorem: According to the Remainder Theorem, a polynomial [tex]f(x)[/tex] divided by [tex](x - a)[/tex] gives a remainder of [tex]f(a)[/tex].
Polynomial division and substitution: When evaluating using the roots of the divisor, [tex]x = 0, x = 1, x = -1[/tex]:
- For [tex]x = 0[/tex], substitute 0 in [tex]f(x)[/tex]: [tex]f(0) = 0^{83} + 0^{65} + \cdots + 0^5 = 0[/tex]
- For [tex]x = 1[/tex], substitute 1 in [tex]f(x)[/tex]: [tex]f(1) = 1^{83} + 1^{65} + \cdots + 1^5 = 7[/tex]
- For [tex]x = -1[/tex], substitute -1 in [tex]f(x)[/tex]: [tex]f(-1) = (-1)^{83} + (-1)^{65} + \dots + (-1)^5[/tex]
- This equals to: [tex]-1 - 1 - 1 - 1 + 1 - 1 - 1 = -5[/tex]
Finding the remainder: The remainder polynomial will be a degree less than [tex]x^5 - x^3[/tex], which could be [tex]x^3[/tex] multiplied by a linear polynomial expression (since the divisor is degree 5).
Combining the remainder values:
- The remainder, when combined, could form [tex]x^3 (x + 6)[/tex], since it meets the properties needed from considering the roots:
- [tex]f(1) = 7[/tex] and [tex]f(-1) = -5[/tex] implies an adjustment linear degree coefficient.
So the correct answer is (D) [tex]x^3(x + 6)[/tex].
