Explanation:
To find the remainder when f(x) = x^6 + 5x^3 - 3 is divided by x + 3, we can use synthetic division. The process of synthetic division involves the following steps:
1. Write down the coefficients of the polynomial in descending order of degree.
2. Bring down the first coefficient (in this case, 1) as the first entry in the division bar.
3. Multiply the divisor (x + 3) by the number in the division bar and write the result underneath the next coefficient.
4. Add the two values in the column to get the next entry in the division bar.
Repeat steps 3 and 4 until all coefficients have been processed.
Using these steps, we get:
-3 | 1 0 0 5 0 0 -3
| -3 9 -42 141 -423
-----------------------
1 -3 9 -37 141 -423
Therefore, the remainder when f(x) is divided by x + 3 is 1x^5 - 3x^4 + 9x^3 - 37x^2 + 141x - 423. We can also express this result in terms of the original polynomial f(x) as:
f(x) = (x + 3)(x^5 - 3x^4 + 9x^3 - 37x^2 + 141x - 423) + 1260
So, the remainder is 1260.