Understanding the Problem
To find the remainder when the polynomial x³ - ax² + 6x - a is divided by (x - a), we can apply the Remainder Theorem. According to this theorem, the remainder of a polynomial f(x) divided by (x - c) is f(c).
Step-by-Step Solution
- Identify the Polynomial:
Let f(x) = x³ - ax² + 6x - a.
- Apply the Remainder Theorem:
We need to substitute x = a into f(x) to find the remainder.
- Substitute x = a:
f(a) = a³ - a(a)² + 6a - a
= a³ - a³ + 6a - a
= 6a - a
= 5a.
Conclusion
The remainder when x³ - ax² + 6x - a is divided by (x - a) is 5a.
Key Takeaways
- Remainder Theorem: Useful for finding remainders without full polynomial long division.
- Substitution: Substitute the value (in this case, a) into the polynomial.
- Final Result: The remainder is 5a, which is dependent on the value of a.
This concise method simplifies the division process and provides an efficient way to determine the remainder.