Question

Use the long division method to find the result when 2, x, cubed, plus, 5, x, squared, minus, 20, x, plus, 42x
3
+5x
2
−20x+4 is divided by x, minus, 2x−2.



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

Let's find the result when \( 2x^3 + 5x^2 - 20x + 42 \) is divided by \( x - 2 \) using long division.

### Use Long Division:

```
2x^2 + 9x + -12
________________________
x - 2 | 2x^3 + 5x^2 - 20x + 42
```

1. **First Step:**
- Divide the first term: \( 2x^3 \div x = 2x^2 \).
- Multiply: \( 2x^2(x - 2) = 2x^3 - 4x^2 \).
- Subtract:
```
2x^2
________________________
x - 2 | 2x^3 + 5x^2 - 20x + 42
- (2x^3 - 4x^2)
________________________
9x^2 - 20x + 42
```

2. **Second Step:**
- Divide the first term: \( 9x^2 \div x = 9x \).
- Multiply: \( 9x(x - 2) = 9x^2 - 18x \).
- Subtract:
```
2x^2 + 9x
________________________
x - 2 | 2x^3 + 5x^2 - 20x + 42
- (2x^3 - 4x^2)
________________________
9x^2 - 20x + 42
- (9x^2 - 18x)
________________________
-2x + 42
```

3. **Third Step:**
- Divide the first term: \( -2x \div x = -2 \).
- Multiply: \( -2(x - 2) = -2x + 4 \).
- Subtract:
```
2x^2 + 9x - 2
________________________
x - 2 | 2x^3 + 5x^2 - 20x + 42
- (2x^3 - 4x^2)
________________________
9x^2 - 20x + 42
- (9x^2 - 18x)
________________________
-2x + 42
- (-2x + 4)
________________________
38
```

### Final Answer:
The result of dividing \( 2x^3 + 5x^2 - 20x + 42 \) by \( x - 2 \) is
\[
2x^2 + 9x - 2 \quad \text{with a remainder} \quad 38.
\]

So the final answer is:
\[
\text{Answer: } 2x^2 + 9x - 2 \quad \text{Remainder: } 38.
\]