(2x^2-5x-3)/(x-3) Long Division

3 min read Jun 16, 2024
(2x^2-5x-3)/(x-3) Long Division

Long Division of Polynomials: (2x^2 - 5x - 3) ÷ (x - 3)

Long division is a powerful technique for dividing polynomials. This method is particularly useful when the divisor is a linear expression, like (x - 3) in our case. Let's dive into the process:

Step 1: Setting up the Division

First, write the dividend (2x^2 - 5x - 3) inside the division symbol and the divisor (x - 3) outside:

          ______
x - 3 | 2x^2 - 5x - 3 

Step 2: Focus on the Leading Terms

We start by looking at the leading terms of the dividend (2x^2) and the divisor (x). Ask yourself: "What do I need to multiply x by to get 2x^2?". The answer is 2x.

Write 2x above the division symbol, aligning it with the x term:

          2x     
x - 3 | 2x^2 - 5x - 3 

Step 3: Multiply and Subtract

Multiply the divisor (x - 3) by the term we just found (2x):

          2x     
x - 3 | 2x^2 - 5x - 3 
          2x^2 - 6x 

Subtract the result from the dividend:

          2x     
x - 3 | 2x^2 - 5x - 3 
          2x^2 - 6x 
          ---------
               x - 3

Step 4: Bring Down the Next Term

Bring down the next term (-3) from the dividend:

          2x     
x - 3 | 2x^2 - 5x - 3 
          2x^2 - 6x 
          ---------
               x - 3

Step 5: Repeat the Process

Now, focus on the new leading term (x) and the leading term of the divisor (x). Ask yourself: "What do I need to multiply x by to get x?". The answer is 1.

Write +1 above the division symbol:

          2x + 1
x - 3 | 2x^2 - 5x - 3 
          2x^2 - 6x 
          ---------
               x - 3

Multiply (x - 3) by 1 and subtract:

          2x + 1
x - 3 | 2x^2 - 5x - 3 
          2x^2 - 6x 
          ---------
               x - 3
               x - 3
               ------
                  0

Step 6: The Result

We have reached a remainder of 0, which means the division is complete.

Therefore, the result of (2x^2 - 5x - 3) ÷ (x - 3) is 2x + 1.

This can also be expressed as:

(2x^2 - 5x - 3) = (x - 3)(2x + 1)

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