Use synthetic division to find the result when is divided by . If there is
remainder, express the result in the form .

Ask by Goodwin Sherman. in the United States

Mar 19,2025

Question

Use synthetic division to find the result when $$ 3 x^{3}+9 x^{2}-5 x-7 $$ is divided by $$ x+3 $$. If there is remainder, express the result in the form $$ q(x)+\frac{r(x)}{b(x)} $$.

UpStudy AI · Accepted Answer

To perform synthetic division of the polynomial $$ 3x^3 + 9x^2 - 5x - 7 $$ by $$ x + 3 $$, we first need to identify the root of the divisor. The root of $$ x + 3 = 0 $$ is $$ x = -3 $$.

Next, we will set up the synthetic division using the coefficients of the polynomial:

- Coefficients of $$ 3x^3 + 9x^2 - 5x - 7 $$ are $$ 3, 9, -5, -7 $$.

Now, we will perform the synthetic division step by step:

1. Write down the coefficients: $$ 3, 9, -5, -7 $$.
2. Write the root $$ -3 $$ to the left.
3. Bring down the leading coefficient $$ 3 $$.
4. Multiply $$ -3 $$ by $$ 3 $$ and add to the next coefficient $$ 9 $$.
5. Repeat the process for the remaining coefficients.

Let's perform these calculations:

- Start with $$ 3 $$:
  - Bring down $$ 3 $$.
  
- Multiply $$ -3 $$ by $$ 3 $$:
  - $$ -3 \times 3 = -9 $$.
  - Add to $$ 9 $$: $$ 9 + (-9) = 0 $$.

- Now, multiply $$ -3 $$ by $$ 0 $$:
  - $$ -3 \times 0 = 0 $$.
  - Add to $$ -5 $$: $$ -5 + 0 = -5 $$.

- Finally, multiply $$ -3 $$ by $$ -5 $$:
  - $$ -3 \times -5 = 15 $$.
  - Add to $$ -7 $$: $$ -7 + 15 = 8 $$.

Now we have the results of the synthetic division:

- The coefficients of the quotient polynomial are $$ 3, 0, -5 $$, which corresponds to $$ 3x^2 - 5 $$.
- The remainder is $$ 8 $$.

Thus, we can express the result of the division as:

$$
q(x) + \frac{r(x)}{b(x)} = 3x^2 - 5 + \frac{8}{x + 3}
$$

So, the final answer is:

$$
3x^2 - 5 + \frac{8}{x + 3}
$$
The result of dividing $$ 3x^3 + 9x^2 - 5x - 7 $$ by $$ x + 3 $$ is $$ 3x^2 - 5 + \frac{8}{x + 3} $$.

Use synthetic division to find the result when is divided by . If there is
remainder, express the result in the form .

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Use synthetic division to find the result when is divided by . If there is remainder, express the result in the form .