If 6x3+2x2−5x+1 6 x 3 + 2 x 2 − 5 x + 1 divides x2−x−1 x 2 − x − 1 , find the remainder.

If $6x^3+2x^2-5x+1$ divides $x^2-x-1$, find the remainder.

Answer Details

We can use polynomial long division to find the quotient and remainder when (6x³ + 2x² - 5x + 1) is divided by (x² - x - 1). The first step is to divide the highest degree term of the dividend (6x³) by the highest degree term of the divisor (x²), which gives us 6x. We then multiply the divisor by this term to get 6x³ - 6x² - 6x, which we subtract from the dividend to get:

6x^3 + 2x^2 - 5x + 1
-(6x^3 - 6x^2 - 6x)
_____________________
     8x^2 + x + 1

We repeat the process with the new polynomial (8x² + x + 1) as the dividend, and the same divisor (x² - x - 1). Dividing the highest degree term (8x²) by the highest degree term of the divisor (x²) gives us 8, so we multiply the divisor by 8 to get 8x² - 8x - 8, which we subtract from the dividend to get:

8x^2 + x + 1
-(8x^2 - 8x - 8)
_________________
        9x + 9

The remainder is 9x + 9. Therefore, the correct answer is 9x + 9.