How do you divide #( -x^3+ 4x^2+8x-7 )/(x - 2 )#?

Utilizing the divisor as a factor in the numerator is one method.

To divide (-x^3 + 4x^2 + 8x - 7) by (x - 2), you can use long division or synthetic division. Here is the step-by-step process using long division:

1. Divide the first term of the numerator (-x^3) by the first term of the denominator (x). The result is -x^2.
2. Multiply the entire denominator (x - 2) by the result obtained in step 1 (-x^2), and write the product below the numerator. -x^2 * (x - 2) = -x^3 + 2x^2
3. Subtract the product obtained in step 2 from the numerator (-x^3 + 4x^2 + 8x - 7) by subtracting term by term. (-x^3 + 4x^2 + 8x - 7) - (-x^3 + 2x^2) = 2x^2 + 8x - 7
4. Bring down the next term from the numerator (-7) and append it to the result obtained in step 3 (2x^2 + 8x - 7).
5. Divide the first term of the new expression (2x^2) by the first term of the denominator (x). The result is 2x.
6. Multiply the entire denominator (x - 2) by the result obtained in step 5 (2x), and write the product below the new expression. 2x * (x - 2) = 2x^2 - 4x
7. Subtract the product obtained in step 6 from the new expression (2x^2 + 8x - 7) by subtracting term by term. (2x^2 + 8x - 7) - (2x^2 - 4x) = 12x - 7
8. Since there are no more terms in the numerator, the division is complete. The quotient is -x^2 + 2x + 2 with a remainder of (12x - 7)/(x - 2).

Therefore, the division of (-x^3 + 4x^2 + 8x - 7) by (x - 2) is equal to -x^2 + 2x + 2 with a remainder of (12x - 7)/(x - 2).