Question

Use the long division method to find the result when 3, x, cubed, minus, 7, x, squared, minus, 20, x, minus, 103x
3
−7x
2
−20x−10 is divided by x, plus, 1x+1.



Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 12, x, squared, plus, 27, x, plus, 27
4x
3
+12x
2
+27x+27 is divided by
2, x, plus, 3
2x+3.
Use Long Division:
Use Long Division:


2, x, squared
2x
2
plus, 3, x
+3x
plus, 9
+9


2, x
2x
plus, 3
+3
right parenthesis
)
4, x, cubed
4x
3
plus, 12, x, squared
+12x
2
plus, 27, x
+27x
plus, 27
+27



start superscript, minus, end superscript, 4, x, cubed
−
4x
3
plus, start superscript, minus, end superscript, 6, x, squared
+
−
6x
2






6, x, squared
6x
2
plus, 27, x
+27x
plus, 27
+27




start superscript, minus, end superscript, 6, x, squared
−
6x
2
plus, start superscript, minus, end superscript, 9, x
+
−
9x






18, x
18x
plus, 27
+27





start superscript, minus, end superscript, 18, x
−
18x
plus, start superscript, minus, end superscript, 27
+
−
27






0
0

Answer: , 2, x, squared, plus, 3, x, plus, 9
Answer: 2x
2
+3x+9

Answer 1

To perform long division on \(3x^3 - 7x^2 - 20x - 103\) divided by \(x + 1\), we will follow the same method as in your example. Here is the step-by-step long division process:

1. **Set up the division**: We divide \(3x^3 - 7x^2 - 20x - 103\) by \(x + 1\).

```
3x^2 - 10x + 10
______________________
x + 1 | 3x^3 - 7x^2 - 20x - 103
```

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

3. **Multiply and subtract**: Multiply \(3x^2\) by \( (x + 1) \) to get \(3x^3 + 3x^2\). Subtract this from \(3x^3 - 7x^2\).

```
3x^2 - 10x + 10
______________________
x + 1 | 3x^3 - 7x^2 - 20x - 103
-(3x^3 + 3x^2)
-------------------
-10x^2 - 20x
```

4. **Bring down the next term**: Now we have \(-10x^2 - 20x\) from the previous subtraction.

5. **Repeat the process**: Divide \(-10x^2\) by \(x\) to get \(-10x\).

6. **Multiply and subtract**: Multiply \(-10x\) by \( (x + 1) \) to get \(-10x^2 - 10x\). Subtract from the current polynomial.

```
3x^2 - 10x + 10
______________________
x + 1 | 3x^3 - 7x^2 - 20x - 103
-(3x^3 + 3x^2)
-------------------
-10x^2 - 20x
-(-10x^2 - 10x)
-------------------
-10x - 103
```

7. **Bring down the next term**: Now we have \(-10x - 103\).

8. **Repeat the process**: Divide \(-10x\) by \(x\) to get \(-10\).

9. **Multiply and subtract**: Multiply \(-10\) by \( (x + 1) \) to get \(-10x - 10\). Subtract from \(-10x - 103\).

```
3x^2 - 10x + 10
______________________
x + 1 | 3x^3 - 7x^2 - 20x - 103
-(3x^3 + 3x^2)
-------------------
-10x^2 - 20x
-(-10x^2 - 10x)
-------------------
-10x - 103
-(-10x - 10)
-------------------
-93
```

10. **The remainder**: We have a remainder of \(-93\).

So the final result of the division \(3x^3 - 7x^2 - 20x - 103\) divided by \(x + 1\) is:

**Answer**: \(3x^2 - 10x - 10\) with a remainder of \(-93\).

Thus, we can express the result as:
\[
3x^2 - 10x - 10 + \frac{-93}{x+1}
\]

This means that the quotient is \(3x^2 - 10x - 10\) and the remainder is \(-93\).