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

Ask by Norris Fleming. in the United States

Nov 12,2024

Question

Use synthetic division to find the result when $$ 2 x^{3}+15 x^{2}+27 x+10 $$ is divided by $$ x+5 $$. If there is a 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 $$ 2x^3 + 15x^2 + 27x + 10 $$ by $$ x + 5 $$, we first rewrite $$ x + 5 $$ in the form $$ x - c $$ where $$ c = -5 $$.

We will use the coefficients of the polynomial $$ 2, 15, 27, 10 $$ for synthetic division. The steps are as follows:

1. Write down the coefficients: $$ 2, 15, 27, 10 $$.
2. Write $$ -5 $$ to the left (the root of $$ x + 5 = 0 $$).
3. Bring down the leading coefficient (the first coefficient).

The synthetic division setup looks like this:

$$
\begin{array}{r|rrrr}
-5 & 2 & 15 & 27 & 10 \
   &   & -10 & -25 & -10 \
\hline
   & 2 & 5 & 2 & 0 \
\end{array}
$$

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

- Bring down the $$ 2 $$.
- Multiply $$ -5 $$ by $$ 2 $$ to get $$ -10 $$ and add it to $$ 15 $$ to get $$ 5 $$.
- Multiply $$ -5 $$ by $$ 5 $$ to get $$ -25 $$ and add it to $$ 27 $$ to get $$ 2 $$.
- Multiply $$ -5 $$ by $$ 2 $$ to get $$ -10 $$ and add it to $$ 10 $$ to get $$ 0 $$.

The final row gives us the coefficients of the quotient polynomial and the remainder:

- The quotient is $$ 2x^2 + 5x + 2 $$.
- The remainder is $$ 0 $$.

Since the remainder is $$ 0 $$, we can express the result of the division as:

$$
q(x) + \frac{r(x)}{b(x)} = 2x^2 + 5x + 2 + \frac{0}{x + 5} = 2x^2 + 5x + 2.
$$

Thus, the result of dividing $$ 2x^3 + 15x^2 + 27x + 10 $$ by $$ x + 5 $$ is:

$$
\boxed{2x^2 + 5x + 2}.
$$
The result of dividing $$ 2x^3 + 15x^2 + 27x + 10 $$ by $$ x + 5 $$ is $$ 2x^2 + 5x + 2 $$.

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