Question

use synthetic division to simplify \\(\frac{6x^3 - 6x^2 + 12}{x + 1}\\).

write your answer in the form \\(q(x) + \frac{r}{d(x)}), where \\(q(x)\\) is a polynomial, \\(r\\) is an integer, and \\(d(x)\\) is a linear polynomial. simplify any fractions.

Explanation:

division
Bring down 6. Multiply by $-1$: $-6$. Add to next coefficient: $-6 + (-6) = -12$. Multiply by $-1$: $12$. Add to next coefficient: $0 + 12 = 12$. Multiply by $-1$: $-12$. Add to last coefficient: $12 + (-12) = 0$.

Step4: Form quotient and remainder

Quotient $q(x)=6x^2 -12x +12$, remainder $r=0$.

Step5: Write final form

$q(x) + \frac{r}{d(x)} = 6x^2 -12x +12 + \frac{0}{x+1}$.

Answer:

division
Bring down 6. Multiply by $-1$: $-6$. Add to next coefficient: $-6 + (-6) = -12$. Multiply by $-1$: $12$. Add to next coefficient: $0 + 12 = 12$. Multiply by $-1$: $-12$. Add to last coefficient: $12 + (-12) = 0$.

Step4: Form quotient and remainder

Quotient $q(x)=6x^2 -12x +12$, remainder $r=0$.

Step5: Write final form

$q(x) + \frac{r}{d(x)} = 6x^2 -12x +12 + \frac{0}{x+1}$.