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

Ask by Wright Ortega. in the United States

Nov 09,2024

Question

Use synthetic division to find the result when $$ 4 x^{4}+25 x^{3}+2 x^{2}-25 x-8 $$ is divided by $$ x+6 $$. 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 $$ 4x^4 + 25x^3 + 2x^2 - 25x - 8 $$ by $$ x + 6 $$, we first note that we can use $$ -6 $$ as the value for synthetic division.

We will set up the synthetic division as follows:

1. Write down the coefficients of the polynomial:
   - The coefficients are $$ 4, 25, 2, -25, -8 $$.

2. Set up the synthetic division with $$ -6 $$:
   $$
   \begin{array}{r|rrrrr}
   -6 & 4 & 25 & 2 & -25 & -8 \
      &   & -24 & -6 & 24 & 6 \
   \hline
      & 4 & 1 & -4 & -1 & -2 \
   \end{array}
   $$

3. Perform the synthetic division:
   - Bring down the leading coefficient $$ 4 $$.
   - Multiply $$ 4 $$ by $$ -6 $$ to get $$ -24 $$ and add it to $$ 25 $$ to get $$ 1 $$.
   - Multiply $$ 1 $$ by $$ -6 $$ to get $$ -6 $$ and add it to $$ 2 $$ to get $$ -4 $$.
   - Multiply $$ -4 $$ by $$ -6 $$ to get $$ 24 $$ and add it to $$ -25 $$ to get $$ -1 $$.
   - Multiply $$ -1 $$ by $$ -6 $$ to get $$ 6 $$ and add it to $$ -8 $$ to get $$ -2 $$.

4. The result of the synthetic division gives us the coefficients of the quotient and the remainder:
   - The quotient is $$ 4x^3 + 1x^2 - 4x - 1 $$.
   - The remainder is $$ -2 $$.

Thus, we can express the result of the division as:
$$
q(x) + \frac{r(x)}{b(x)} = 4x^3 + x^2 - 4x - 1 + \frac{-2}{x + 6}
$$

Therefore, the final result is:
$$
4x^3 + x^2 - 4x - 1 - \frac{2}{x + 6}
$$
The result of the division is $$ 4x^3 + x^2 - 4x - 1 - \frac{2}{x + 6} $$.

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

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

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