Question

Divide using synthetic division: (k⁴ - 5k³ - k² + 13k - 12) ÷ (k - 6).

Answer 1

The synthetic division of the polynomial k⁴ - 5k³ - k² + 13k - 12 by k - 6 yields a quotient of k³ + k² - 7k + 55 with a remainder of -342.

Step-by-step explanation:

To divide the polynomial k⁴ - 5k³ - k² + 13k - 12 by k - 6 using synthetic division, we first write down the coefficients of the polynomial, which are 1, -5, -1, 13, and -12.

We then write the zero of the divisor k - 6, which is k = 6, to the left of a vertical bar and the coefficients to the right. We follow the steps of synthetic division, beginning by bringing down the leading coefficient.

1. Write down the coefficients of the polynomial: 1, -5, -1, 13, -12.
2. Write the zero of the divisor, which is 6, to the left of the bar.
3. Bring down the leading coefficient, which is 1.
4. Multiply the zero by the number just written below the line, then write this product under the next coefficient.
5. Add the numbers in each column and write the sum below the line.
6. Repeat steps 4-5 until all coefficients have been used.

The numbers at the bottom are the coefficients of the answer, so the result is k³ + k² - 7k + 55 with a remainder of -342. Therefore, the quotient is k³ + k² - 7k + 55 and the remainder is -342.