Question

use synthetic division to divide $x^3 + 6x^2 + 10x + 11$ by $x + 1$

1 6 10 11

the quotient is:
the remainder is:
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Explanation:

Step1: Find root of divisor

To use synthetic division, solve $x+1=0$, so $x=-1$.

Step2: Bring down leading coefficient

Bring down the first coefficient, $1$.

Step3: Multiply and add (1st column)

Multiply $1 \times (-1) = -1$, add to $6$: $6 + (-1) = 5$

Step4: Multiply and add (2nd column)

Multiply $5 \times (-1) = -5$, add to $10$: $10 + (-5) = 5$

Step5: Multiply and add (3rd column)

Multiply $5 \times (-1) = -5$, add to $11$: $11 + (-5) = 6$

Filling the synthetic division table:
Top left box: $-1$
Middle row boxes (left to right): $-1$, $-5$, $-5$
Bottom row boxes (left to right): $1$, $5$, $5$, $6$

Step6: Identify quotient and remainder

The quotient uses the first 3 bottom values as coefficients of $x^2, x, constant$. The last value is the remainder.

Answer:

Synthetic division table filled:
$-1$ $\bigg|$ $1$ $\quad$ $6$ $\quad$ $10$ $\quad$ $11$
$\quad$ $\bigg|$ $\quad$ $-1$ $\quad$ $-5$ $\quad$ $-5$
$\quad$ $\overline{\bigg|}$ $\overline{1}$ $\quad$ $\overline{5}$ $\quad$ $\overline{5}$ $\quad$ $\overline{6}$

The quotient is: $x^2 + 5x + 5$
The remainder is: $6$