Question

Long division x-2/2x^2-5x-7

Answer 1

surely you cannot believe all that rubbish.
a 1st-degree polynomial divided by a second-degree polynomial cannot yield a first degree as a result!
x^1/x^2 = 1/x

In fact, (x-2)/(2x^2-5x-7)
= 1/(2x) + 1/(4x^2) + 19(8x^3) + ...

So the user probably meant
(2x^2-5x-7)/(x-2) = 2x-1 - 9/(x-2)

Answer 2

To divide (x-2) by (2x^2 - 5x - 7), you start by setting up the division in the long division format.

```
_______________________
2x^2 - 5x - 7 | x - 2
```

First, divide the first term x of the dividend (x - 2) by the first term 2x^2 of the divisor (2x^2 - 5x - 7). Then write the result above the horizontal line.

2x^2 / 2x^2 = x

```
_______________________
2x^2 - 5x - 7 | x - 2
x
```

Next, multiply the divisor (2x^2 - 5x - 7) by the result obtained (x) and write the product below the dividend.

x * (2x^2 - 5x - 7) = 2x^3 - 5x^2 - 7x

```
_______________________
2x^2 - 5x - 7 | x - 2
-2x^3 + 5x^2 + 7x
```

Now, subtract the product from the dividend (x - 2)

(x - 2) - (-2x^3 + 5x^2 + 7x) = -2x^3 + 5x^2 + 8x - 2

```
_______________________
2x^2 - 5x - 7 | x - 2
-2x^3 + 5x^2 + 7x
_______________________
8x - 2
```

Now, bring down the next term (-2) from the original dividend (x - 2) and divide it by the original divisor (2x^2 - 5x - 7).

8x / 2x^2 = 4x

```
_______________________
2x^2 - 5x - 7 | x - 2
-2x^3 + 5x^2 + 7x
_______________________
8x - 2
8x
```

Multiply the divisor (2x^2 - 5x - 7) by the result obtained (4x) and write the product below the previous subtraction.

4x * (2x^2 - 5x - 7) = 8x^3 - 20x^2 - 28x

```
_______________________
2x^2 - 5x - 7 | x - 2
-2x^3 + 5x^2 + 7x
_______________________
8x - 2
8x + (-8x^3 + 20x^2 + 28x)
```

Subtract the product from the previous subtraction.

(8x - 2) - (8x^3 - 20x^2 - 28x) = -8x^3 + 20x^2 + 36x - 2

```
_______________________
2x^2 - 5x - 7 | x - 2
-2x^3 + 5x^2 + 7x
_______________________
8x - 2
8x + (-8x^3 + 20x^2 + 28x)
_______________________
36x - 2
```

Now, bring down any remaining terms from the original dividend (in this case, there are no more terms), and the division is complete.

Therefore, the result of the division of (x - 2) by (2x^2 - 5x - 7) is:

Quotient: x + 4

Remainder: 36x - 2

Answer 3

Apologies for the mistake in my previous response. You are correct that the division of a first-degree polynomial by a second-degree polynomial cannot yield a first-degree result.

The correct division should be:

```
______________________
x - 2 | 2x^2 - 5x - 7
```

To divide, start by dividing the first term 2x^2 of the dividend (2x^2 - 5x - 7) by the first term x of the divisor (x - 2). Then write the result above the horizontal line.

2x^2 / x = 2x

```
______________________
x - 2 | 2x^2 - 5x - 7
2x
```

Next, multiply the divisor (x - 2) by the result obtained (2x) and write the product below the dividend.

2x * (x - 2) = 2x^2 - 4x

```
______________________
x - 2 | 2x^2 - 5x - 7
2x
______________________
- 4x
```

Now, subtract the product from the dividend (2x^2 - 5x - 7).

(2x^2 - 5x - 7) - (2x^2 - 4x) = -5x + 7x - 7 = 2x - 7

```
______________________
x - 2 | 2x^2 - 5x - 7
2x
______________________
- 4x
__________
2x - 7
```

Now, bring down any remaining terms from the original dividend, which is -7 in this case.

The remainder is -7.

Therefore, the result of the division of (2x^2 - 5x - 7) by (x - 2) is:

Quotient: 2x

Remainder: -7