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(10x^3+5x^2-1)/(2x^3-4x^2-x+2) Long Division

4 min read Jun 16, 2024
(10x^3+5x^2-1)/(2x^3-4x^2-x+2) Long Division

Long Division of Polynomials: (10x^3+5x^2-1)/(2x^3-4x^2-x+2)

This article will guide you through the process of performing long division with the polynomials (10x³ + 5x² - 1) and (2x³ - 4x² - x + 2).

Setting Up the Problem

  1. Write the problem in long division format:
         ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1

Performing the Division

  1. Divide the leading term of the dividend (10x³) by the leading term of the divisor (2x³). This gives us 5. Write this above the dividend.

         5        
     ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1

  2. Multiply the divisor (2x³ - 4x² - x + 2) by the quotient term (5). This gives us 10x³ - 20x² - 5x + 10. Write this below the dividend.

         5        
     ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
             10x³ - 20x² - 5x + 10

  3. Subtract the result from the dividend. Remember to change the signs of the terms you are subtracting.

         5        
     ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
             10x³ - 20x² - 5x + 10
             -----------------
                   25x² + 5x - 11

  4. Bring down the next term of the dividend (-1).

         5        
     ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
             10x³ - 20x² - 5x + 10
             -----------------
                   25x² + 5x - 11

  5. Repeat steps 1-4 until the degree of the remainder is less than the degree of the divisor.

    • Divide the leading term of the new dividend (25x²) by the leading term of the divisor (2x³). This gives us 25/(2x).
    • Multiply the divisor (2x³ - 4x² - x + 2) by 25/(2x). This gives us (25x² - 50x - 25/2 + 25).
    • Subtract this result from the previous remainder.
    • Bring down the next term of the dividend (-1).
    • Continue this process until the remainder is a constant term.
         5 + 25/(2x)      
     ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
             10x³ - 20x² - 5x + 10
             -----------------
                   25x² + 5x - 11 
                   25x² - 50x - 25/2 + 25
                   ------------------------
                           55x + 14/2

Result

The final result is: (10x³ + 5x² - 1)/(2x³ - 4x² - x + 2) = 5 + 25/(2x) + (55x + 14/2) / (2x³ - 4x² - x + 2)

This can also be written as: 5 + 25/(2x) + (55x + 7)/(2x³ - 4x² - x + 2)

Remember to always check your answer by multiplying the quotient and the divisor and adding the remainder. This should equal the original dividend.

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