(4x2−13x−5)÷(x−2)=

3 min read Jun 16, 2024

Solving Polynomial Division: (4x²−13x−5) ÷ (x−2)

This article will guide you through the process of dividing the polynomial (4x²−13x−5) by (x−2). We'll use the long division method to achieve this.

1. Set Up the Division

Begin by writing the problem in a long division format:

          _______
x - 2 | 4x² - 13x - 5

2. Divide the Leading Terms

Focus on the leading term of the divisor (x) and the leading term of the dividend (4x²).
Ask yourself: "What do I need to multiply x by to get 4x²?".
The answer is 4x.
Write 4x above the dividend.

          4x ______
x - 2 | 4x² - 13x - 5

3. Multiply and Subtract

Multiply (x - 2) by 4x: 4x(x - 2) = 4x² - 8x
Write this result below the dividend.
Subtract the entire expression from the dividend.

          4x ______
x - 2 | 4x² - 13x - 5 
          -(4x² - 8x) 
          ----------
                -5x - 5

4. Bring Down the Next Term

Bring down the next term of the dividend (-5).

          4x ______
x - 2 | 4x² - 13x - 5 
          -(4x² - 8x) 
          ----------
                -5x - 5

5. Repeat the Process

Now, focus on the new leading term (-5x) and the leading term of the divisor (x).
Ask: "What do I need to multiply x by to get -5x?".
The answer is -5.
Write -5 next to 4x above the dividend.

          4x - 5 ______
x - 2 | 4x² - 13x - 5 
          -(4x² - 8x) 
          ----------
                -5x - 5

Multiply (x - 2) by -5: -5(x - 2) = -5x + 10
Write this result below the last line.
Subtract the entire expression.

          4x - 5 ______
x - 2 | 4x² - 13x - 5 
          -(4x² - 8x) 
          ----------
                -5x - 5 
                -(-5x + 10)
                ----------
                        -15

6. The Remainder

Since the degree of -15 is less than the degree of (x - 2), the division process is complete.

Therefore, (4x²−13x−5) ÷ (x−2) = 4x - 5 - 15/(x - 2).