(3x2−14x−5)÷(x−5)=

5 min read Jun 16, 2024

Solving the Division: (3x² - 14x - 5) ÷ (x - 5)

This problem involves polynomial division. There are two main ways to approach this:

This method is similar to long division with numbers.

Steps:

Set up the division: Write the dividend (3x² - 14x - 5) inside the division symbol and the divisor (x - 5) outside.
```
     ___________
x - 5 | 3x² - 14x - 5 
```
Divide the leading terms: Divide the leading term of the dividend (3x²) by the leading term of the divisor (x). This gives us 3x. Write this above the line.
```
     3x       
x - 5 | 3x² - 14x - 5 
```
Multiply the quotient by the divisor: Multiply the quotient (3x) by the divisor (x - 5) and write the result below the dividend.
```
     3x       
x - 5 | 3x² - 14x - 5 
        3x² - 15x
```

Subtract: Subtract the result from the dividend.

     3x       
x - 5 | 3x² - 14x - 5 
        3x² - 15x
        -------
              x - 5

Bring down the next term: Bring down the next term from the dividend (-5).

     3x       
x - 5 | 3x² - 14x - 5 
        3x² - 15x
        -------
              x - 5

Repeat steps 2-5: Divide the new leading term (x) by the leading term of the divisor (x). This gives us 1. Write it above the line.
```
     3x + 1
x - 5 | 3x² - 14x - 5 
        3x² - 15x
        -------
              x - 5
              x - 5 
```

Subtract: Subtract the result.

     3x + 1
x - 5 | 3x² - 14x - 5 
        3x² - 15x
        -------
              x - 5
              x - 5 
              -----
               0

Therefore, (3x² - 14x - 5) ÷ (x - 5) = 3x + 1.

This method is a shortcut for long division when the divisor is of the form (x - a).

Steps:

Write the coefficients: Write the coefficients of the dividend (3, -14, -5) and the constant term of the divisor (5).
```
    5 | 3  -14  -5
```
Bring down the first coefficient: Bring down the first coefficient (3) below the line.
```
    5 | 3  -14  -5
        3
```
Multiply and add: Multiply the brought-down coefficient (3) by the constant term of the divisor (5) and write the product (15) below the next coefficient (-14). Add the two numbers (-14 + 15 = 1).
```
    5 | 3  -14  -5
        3   15
        -------
        3    1
```
Repeat step 3: Multiply the new number (1) by the constant term of the divisor (5) and write the product (5) below the next coefficient (-5). Add the two numbers (-5 + 5 = 0).
```
    5 | 3  -14  -5
        3   15   5
        -------
        3    1    0
```
Interpret the result: The last number (0) is the remainder. The other numbers (3, 1) are the coefficients of the quotient.

Therefore, the quotient is 3x + 1 and the remainder is 0. This confirms that (3x² - 14x - 5) ÷ (x - 5) = 3x + 1.