Question

polynomial long division (level 3)
score: 1/2 penalty: 0.25 off
question
use the long division method to find the result when $x^3 - 5x^2 + 6x - 12$ is divided by $x - 1$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.
answer attempt 1 out of 2

Explanation:

Step1: Divide leading terms

$\frac{x^3}{x} = x^2$

Step2: Multiply divisor by $x^2$

$x^2(x - 1) = x^3 - x^2$

Step3: Subtract from dividend

$(x^3 - 5x^2 + 6x - 12) - (x^3 - x^2) = -4x^2 + 6x - 12$

Step4: Divide new leading term

$\frac{-4x^2}{x} = -4x$

Step5: Multiply divisor by $-4x$

$-4x(x - 1) = -4x^2 + 4x$

Step6: Subtract again

$(-4x^2 + 6x - 12) - (-4x^2 + 4x) = 2x - 12$

Step7: Divide next leading term

$\frac{2x}{x} = 2$

Step8: Multiply divisor by 2

$2(x - 1) = 2x - 2$

Step9: Subtract for remainder

$(2x - 12) - (2x - 2) = -10$

Step10: Combine quotient and remainder

$q(x) + \frac{r(x)}{b(x)} = x^2 - 4x + 2 + \frac{-10}{x - 1}$

Answer:

$x^2 - 4x + 2 - \frac{10}{x - 1}$