Question

Divide x^3+6x^2-6x+8/x-7 using synthetic division

Answer 1

Synthetic division is used to divide the polynomial x^3+6x^2-6x+8 by x-7, resulting in a quotient of x^2+13x+85 and a remainder of 593. The process involves using the root of the divisor, multiplying, and adding the coefficients in a systematic manner.

Step-by-step explanation:

To divide x^3+6x^2-6x+8 by x-7 using synthetic division, follow these steps:

1. Write down the coefficients of the polynomial x^3+6x^2-6x+8, which are 1, 6, -6, and 8.
2. Use the root of the divisor, which in this case is 7 (since we're dividing by x-7), and write it to the left of the coefficients.
3. Bring down the leading coefficient (which is 1) to the bottom row.
4. Multiply this number by 7 and write the result under the second coefficient. Add these two numbers together and write the sum in the bottom row.
5. Repeat the multiply and add process with the new number in the bottom row and the next coefficient (-6), and then again with the number obtained and the last coefficient (8).
6. The numbers at the bottom row will represent the coefficients of the quotient polynomial. The remainder, if any, is the last number.

After performing synthetic division, we'll find that the quotient is x^2+13x+85 and the remainder is 593. The result can be expressed as:

x^2+13x+85 + 593/(x-7)

To ensure the answer is reasonable, check that:

• The degree of the quotient polynomial is one less than the original polynomial.
• When you multiply the quotient polynomial by x-7 and add the remainder, you should get the original polynomial back.