(2x4 – 3x3 – 3x2 + 7x – 3) ÷ (x2 – 2x + 1)

Jasonbradley

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Jun 16, 2024

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Solving Polynomial Division: (2x⁴ – 3x³ – 3x² + 7x – 3) ÷ (x² – 2x + 1)

This article will guide you through the process of dividing the polynomial (2x⁴ – 3x³ – 3x² + 7x – 3) by (x² – 2x + 1) using long division.

Understanding Long Division of Polynomials

Long division of polynomials follows a similar principle to numerical long division. We aim to find a quotient polynomial that, when multiplied by the divisor, results in the dividend.

Step-by-Step Solution

1. Set up the problem:

       ____________
   x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 
   

2. Focus on the leading terms:

   • Divide the leading term of the dividend (2x⁴) by the leading term of the divisor (x²): 2x⁴ / x² = 2x². This is the first term of the quotient.

   • Write 2x² above the dividend.

   • Multiply the divisor (x² – 2x + 1) by 2x²: 2x² (x² – 2x + 1) = 2x⁴ – 4x³ + 2x²

   • Subtract this product from the dividend:

       2x²
       ____________
     

   x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) -------------------- x³ - 5x² + 7x

3. Repeat the process:

   • Bring down the next term of the dividend (7x).

   • Focus on the leading terms again: x³ / x² = x. This is the next term of the quotient.

   • Write x above the dividend, next to 2x².

   • Multiply the divisor by x: x (x² – 2x + 1) = x³ – 2x² + x

   • Subtract this product from the current result:

       2x² + x
       ____________
     

   x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) -------------------- x³ - 5x² + 7x -(x³ - 2x² + x) -------------------- -3x² + 6x - 3

4. Final step:

   • Bring down the last term of the dividend (-3).

   • Focus on the leading terms: -3x² / x² = -3. This is the final term of the quotient.

   • Write -3 above the dividend, next to x.

   • Multiply the divisor by -3: -3 (x² – 2x + 1) = -3x² + 6x - 3

   • Subtract this product from the current result:

       2x² + x - 3
       ____________
     

   x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) -------------------- x³ - 5x² + 7x -(x³ - 2x² + x) -------------------- -3x² + 6x - 3 -(-3x² + 6x - 3) -------------------- 0

Conclusion

The division is complete, as the remainder is 0. Therefore:

(2x⁴ – 3x³ – 3x² + 7x – 3) ÷ (x² – 2x + 1) = 2x² + x - 3