(x^2+5x-14)/(x-2) Long Division

3 min read Jun 17, 2024

Long Division of Polynomials: (x² + 5x - 14) / (x - 2)

Long division with polynomials follows a similar process to long division with numbers. Let's divide (x² + 5x - 14) by (x - 2).

Set up the division:
```
     _______
x - 2 | x² + 5x - 14
```
Divide the leading terms:
- The leading term of the divisor (x - 2) is 'x'.
- The leading term of the dividend (x² + 5x - 14) is 'x²'.
- x² / x = x, which goes above the line.
```
     x______
x - 2 | x² + 5x - 14
```
Multiply the quotient (x) by the divisor (x - 2):
- x * (x - 2) = x² - 2x. Write this result below the dividend.
```
     x______
x - 2 | x² + 5x - 14
       x² - 2x 
```
Subtract:
- Subtract the result (x² - 2x) from the dividend. Remember to change the signs when subtracting.
```
     x______
x - 2 | x² + 5x - 14
       x² - 2x 
       -------
           7x - 14 
```

Bring down the next term:

     x______
x - 2 | x² + 5x - 14
       x² - 2x 
       -------
           7x - 14

Repeat steps 2-5:
- Divide the leading term of the new dividend (7x) by the leading term of the divisor (x). This gives us 7.
- Multiply 7 by the divisor (x - 2) to get 7x - 14.
- Subtract this result from the previous line.
```
     x + 7____
x - 2 | x² + 5x - 14
       x² - 2x 
       -------
           7x - 14
           7x - 14
           -------
               0 
```
The remainder is 0:
- Since the remainder is 0, the division is complete.

Therefore, (x² + 5x - 14) / (x - 2) = x + 7.

This means that (x - 2) is a factor of (x² + 5x - 14).