Question

use the long division method to find the result when 3x^3+x^2-10x-9 is divided by x+1 if there is a remainder express the result in the form q(x)+r(x)/b(x)
please dont get another wrong answer

Answer 1

To perform long division of the polynomial \(3x^3 + x^2 - 10x - 9\) by \(x + 1\):

1. **Set up the long division**:
\[
\begin{array}{r|l}
& 3x^2 - 2x - 8 \\
\hline
x+1 & 3x^3 + x^2 - 10x - 9 \\
\end{array}
\]

2. **Divide the first term**:
Divide \(3x^3\) by \(x\) to get \(3x^2\).

3. **Multiply and subtract**:
Multiply \(3x^2\) by \(x + 1\):
\[
3x^2(x + 1) = 3x^3 + 3x^2
\]
Now subtract this from the original polynomial:
\[
(3x^3 + x^2 - 10x - 9) - (3x^3 + 3x^2) = (x^2 - 3x^2) - 10x - 9 = -2x^2 - 10x - 9
\]

4. **Repeat the process**:
Now divide \(-2x^2\) by \(x\) to get \(-2x\).

5. **Multiply and subtract again**:
Multiply \(-2x\) by \(x + 1\):
\[
-2x(x + 1) = -2x^2 - 2x
\]
Subtract:
\[
(-2x^2 - 10x - 9) - (-2x^2 - 2x) = (-10x + 2x) - 9 = -8x - 9
\]

6. **Repeat again**:
Divide \(-8x\) by \(x\) to get \(-8\).

7. **Multiply and subtract**:
Multiply \(-8\) by \(x + 1\):
\[
-8(x + 1) = -8x - 8
\]
Subtract:
\[
(-8x - 9) - (-8x - 8) = -9 + 8 = -1
\]

Now we have completed the long division and end up with:
- Quotient: \(3x^2 - 2x - 8\)
- Remainder: \(-1\)

Thus, we can express the result as:
\[
\frac{3x^3 + x^2 - 10x - 9}{x + 1} = 3x^2 - 2x - 8 + \frac{-1}{x + 1}
\]

So, the final result is:
\[
q(x) = 3x^2 - 2x - 8,\quad r(x) = -1,\quad b(x) = x + 1
\]

In summary:
\[
3x^3 + x^2 - 10x - 9 = (x + 1)(3x^2 - 2x - 8) - 1
\]