(2x^3+3x^2+5x+9)/(x-20)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial 2x³ + 3x² + 5x + 9 by (x - 20) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division method used for numbers. It involves a series of steps to find the quotient and remainder of the division.

Step-by-Step Process

1. Set up the division problem:

       ___________
   x-20 | 2x³ + 3x² + 5x + 9 
   

2. Divide the leading term of the dividend (2x³) by the leading term of the divisor (x):

   • 2x³ / x = 2x²
   • Write 2x² above the dividend.

       2x²________
   x-20 | 2x³ + 3x² + 5x + 9 
   

3. Multiply the divisor (x - 20) by the term you just wrote (2x²):

   • (x - 20) * 2x² = 2x³ - 40x²

4. Subtract the result from the dividend:

       2x²________
   x-20 | 2x³ + 3x² + 5x + 9 
          -(2x³ - 40x²)
          -----------
                 43x² + 5x
   

5. Bring down the next term of the dividend (5x):

       2x²________
   x-20 | 2x³ + 3x² + 5x + 9 
          -(2x³ - 40x²)
          -----------
                 43x² + 5x 
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (43x²) by the leading term of the divisor (x):
     • 43x² / x = 43x
     • Write 43x above the dividend.
   • Multiply the divisor (x - 20) by 43x:
     • (x - 20) * 43x = 43x² - 860x
   • Subtract the result:

      2x² + 43x_____
     

   x-20 | 2x³ + 3x² + 5x + 9 -(2x³ - 40x²) ----------- 43x² + 5x -(43x² - 860x) ------------- 865x + 9

   * Bring down the next term (9):
   

   2x² + 43x_____
   

   x-20 | 2x³ + 3x² + 5x + 9 -(2x³ - 40x²) ----------- 43x² + 5x -(43x² - 860x) ------------- 865x + 9

7. Repeat steps 2-5 again:

   • Divide the leading term of the new dividend (865x) by the leading term of the divisor (x):
     • 865x / x = 865
     • Write 865 above the dividend.
   • Multiply the divisor (x - 20) by 865:
     • (x - 20) * 865 = 865x - 17300
   • Subtract the result:

      2x² + 43x + 865___
     

   x-20 | 2x³ + 3x² + 5x + 9 -(2x³ - 40x²) ----------- 43x² + 5x -(43x² - 860x) ------------- 865x + 9 -(865x - 17300) ------------- 17309

Conclusion

Therefore, the result of dividing 2x³ + 3x² + 5x + 9 by (x - 20) is:

Quotient: 2x² + 43x + 865 Remainder: 17309

We can express this result as:

(2x³ + 3x² + 5x + 9) / (x - 20) = 2x² + 43x + 865 + 17309 / (x - 20)