%2F(x-3)-long-division.jpeg "(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)