Question

Use the long division method to find the result when 4, x, cubed, minus, x, squared, minus, 13, x, plus, 104x
3
−x
2
−13x+10 is divided by x, minus, 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
+
−

Answer 1

Let's perform long division of the polynomial \(4x^3 - x^2 - 13x + 104\) by \(x - 1\).

### Long Division Steps:

1. **Set up the long division**:

```
_______________________
x - 1 | 4x^3 - x^2 - 13x + 104
```

2. **Divide the first term**:

Divide \(4x^3\) by \(x\):
- Result: \(4x^2\)

Place \(4x^2\) above the long division line:

```
4x^2
_______________________
x - 1 | 4x^3 - x^2 - 13x + 104
```

3. **Multiply and subtract**:

Multiply \(4x^2\) by \(x - 1\):
- Result: \(4x^3 - 4x^2\)

Subtract this from the original polynomial:

```
4x^2
_______________________
x - 1 | 4x^3 - x^2 - 13x + 104
- (4x^3 - 4x^2)
_______________________
3x^2 - 13x + 104
```

4. **Next term**:

Divide \(3x^2\) by \(x\):
- Result: \(3x\)

Add this to the division result:

```
4x^2 + 3x
_______________________
x - 1 | 4x^3 - x^2 - 13x + 104
- (4x^3 - 4x^2)
_______________________
3x^2 - 13x + 104
```

5. **Multiply and subtract**:

Multiply \(3x\) by \(x - 1\):
- Result: \(3x^2 - 3x\)

Subtract:

```
4x^2 + 3x
_______________________
x - 1 | 4x^3 - x^2 - 13x + 104
- (4x^3 - 4x^2)
_______________________
3x^2 - 13x + 104
- (3x^2 - 3x)
_______________________
-10x + 104
```

6. **Next term**:

Divide \(-10x\) by \(x\):
- Result: \(-10\)

Add this to the division result:

```
4x^2 + 3x - 10
_______________________
x - 1 | 4x^3 - x^2 - 13x + 104
- (4x^3 - 4x^2)
_______________________
3x^2 - 13x + 104
- (3x^2 - 3x)
_______________________
-10x + 104
- (-10x + 10)
_______________________
94
```

7. **Final result**:

Since we can't divide further (the degree of \(94\) is less than that of \(x - 1\)), we can write the complete answer. The quotient is:

\[
\boxed{4x^2 + 3x - 10} \quad \text{with a remainder of } \quad 94.
\]

Thus, the answer is:
\[
4x^2 + 3x - 10 + \frac{94}{x - 1}.
\]