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(4x2−13x−5)÷(x−2)

4 min read Jun 16, 2024
(4x2−13x−5)÷(x−2)

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) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division we learned in elementary school, but instead of working with numbers, we work with polynomials. The goal is to find the quotient and remainder when dividing one polynomial by another.

Steps to Divide (4x²−13x−5) by (x−2)

  1. Set up the division: Write the dividend (4x²−13x−5) inside the division symbol and the divisor (x−2) outside.

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

  2. Divide the leading terms: Focus on the leading terms of the dividend (4x²) and the divisor (x).
    Ask: "What do I need to multiply 'x' by to get 4x²?" The answer is 4x. Write 4x above the division symbol.

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

  3. Multiply and subtract: Multiply the divisor (x−2) by 4x and write the result below the dividend.

         4x _______
x - 2 | 4x² - 13x - 5
       4x² - 8x

    Subtract the entire expression (4x² - 8x) from the dividend.

         4x _______
x - 2 | 4x² - 13x - 5
       4x² - 8x 
       -------
           -5x - 5

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

         4x _______
x - 2 | 4x² - 13x - 5
       4x² - 8x 
       -------
           -5x - 5

  5. Repeat steps 2-4: 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 above the division symbol.

         4x - 5 _______
x - 2 | 4x² - 13x - 5
       4x² - 8x 
       -------
           -5x - 5 
           -5x + 10

    Multiply (x - 2) by -5 and subtract the result.

         4x - 5 _______
x - 2 | 4x² - 13x - 5
       4x² - 8x 
       -------
           -5x - 5 
           -5x + 10
           -------
                -15

  6. The remainder: We are left with a remainder of -15.

Conclusion

Therefore, the result of the polynomial division (4x²−13x−5) ÷ (x−2) is:

(4x²−13x−5) ÷ (x−2) = 4x - 5 with a remainder of -15

This can also be written as:

(4x²−13x−5) = (x−2)(4x - 5) - 15

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