Use synthetic division to find the result when $$ x^{4}-x^{3} \pm 4 x^{2}-5 x-3 $$ 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 Submit Answer

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To perform synthetic division of the polynomial $$ x^4 - x^3 \pm 4x^2 - 5x - 3 $$ by $$ x + 1 $$, we first need to identify the value of $$ c $$ for the divisor $$ x + 1 $$. Here, $$ c = -1 $$.

Let's denote the polynomial as $$ P(x) = x^4 - x^3 \pm 4x^2 - 5x - 3 $$. We will use synthetic division with $$ c = -1 $$.

1. Write down the coefficients of $$ P(x) $$:
   - For $$ x^4 $$: 1
   - For $$ x^3 $$: -1
   - For $$ x^2 $$: $$ \pm 4 $$ (we'll consider both cases)
   - For $$ x^1 $$: -5
   - For the constant term: -3

Thus, the coefficients are:
   - Case 1 (using $$ +4 $$): $$ [1, -1, 4, -5, -3] $$
   - Case 2 (using $$ -4 $$): $$ [1, -1, -4, -5, -3] $$

2. Set up the synthetic division:

For Case 1 (using $$ +4 $$):
```
-1 |  1   -1    4   -5   -3
   |       -1    2   -6   11
---------------------------
     1   -2    6  -11    8
```

For Case 2 (using $$ -4 $$):
```
-1 |  1   -1   -4   -5   -3
   |       -1    2    2   3
---------------------------
     1   -2   -2   -3    0
```

3. Interpret the results:
- For Case 1 (using $$ +4 $$):
   - The quotient is $$ x^3 - 2x^2 + 6x - 11 $$ and the remainder is $$ 8 $$.
   - Thus, we can express the result as:
   $$
   P(x) = (x^3 - 2x^2 + 6x - 11) + \frac{8}{x + 1}
   $$

- For Case 2 (using $$ -4 $$):
   - The quotient is $$ x^3 - 2x^2 - 2x - 3 $$ and the remainder is $$ 0 $$.
   - Thus, we can express the result as:
   $$
   P(x) = x^3 - 2x^2 - 2x - 3
   $$

In summary:
- For $$ P(x) = x^4 - x^3 + 4x^2 - 5x - 3 $$:
   $$
   P(x) = (x^3 - 2x^2 + 6x - 11) + \frac{8}{x + 1}
   $$
- For $$ P(x) = x^4 - x^3 - 4x^2 - 5x - 3 $$:
   $$
   P(x) = x^3 - 2x^2 - 2x - 3
   $$
For $$ P(x) = x^4 - x^3 + 4x^2 - 5x - 3 $$, the result is $$ (x^3 - 2x^2 + 6x - 11) + \frac{8}{x + 1} $$. For $$ P(x) = x^4 - x^3 - 4x^2 - 5x - 3 $$, the result is $$ x^3 - 2x^2 - 2x - 3 $$.

Use synthetic division to find the result when \( x^{4}-x^{3} \pm 4 x^{2}-5 x-3 \) 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 Submit Answer

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