Final answer:
Using synthetic division to divide the polynomial by (x - 3), we find that the quotient is (x⁴ - 12x³ + 54x² - 162x + 513) with a remainder, making Option B the correct answer.
Step-by-step explanation:
To find the quotient and remainder using synthetic division for the polynomial (x⁵ - 15x⁴ + 90x³ - 270x² + 405x - 243) divided by (x - 3), we set up the synthetic division as follows:
- Place the coefficients of the polynomial in a row: 1, -15, 90, -270, 405, -243.
- Write the zero of the divisor x - 3, which is 3, to the left of the bar.
- Bring down the leading coefficient (1) to the bottom row.
- Multiply this coefficient by 3 and place the result under the next coefficient.
- Continue this process of adding and multiplying by 3 until you have processed all coefficients.
The result will give us the coefficients of the quotient polynomial plus the remainder. The correct answer to this problem is Option B, which represents the quotient polynomial (x⁴ - 12x³ + 54x² - 162x + 513) and has a remainder when divided by (x - 3).