(4x^3+27x^2+3x+64)/(x+7)

3 min read Jun 16, 2024
(4x^3+27x^2+3x+64)/(x+7)

Polynomial Long Division: (4x^3 + 27x^2 + 3x + 64) / (x + 7)

This article will guide you through the process of dividing the polynomial 4x³ + 27x² + 3x + 64 by x + 7 using long division.

1. Setting Up the Division

First, set up the long division problem like you would with regular numbers:

             __________
x + 7 | 4x³ + 27x² + 3x + 64 

2. Dividing the Leading Terms

  • Focus on the leading terms: Divide the leading term of the dividend (4x³) by the leading term of the divisor (x).
  • Write the result above the line: 4x³ / x = 4x², so write 4x² above the line.
             4x²        
x + 7 | 4x³ + 27x² + 3x + 64 

3. Multiply and Subtract

  • Multiply the divisor by the term you just wrote: (x + 7) * 4x² = 4x³ + 28x²
  • Write the result below the dividend:
  • Subtract the result from the dividend:
             4x²        
x + 7 | 4x³ + 27x² + 3x + 64 
         -(4x³ + 28x²)
              -x² + 3x 

4. Repeat the Process

  • Bring down the next term: Bring down the +3x term.
  • Repeat steps 2 and 3: Divide the new leading term (-x²) by the leading term of the divisor (x).
  • Write the result above the line: -x² / x = -x, so write -x above the line.
  • Multiply the divisor by -x: (x + 7) * -x = -x² - 7x.
  • Subtract the result:
             4x² - x       
x + 7 | 4x³ + 27x² + 3x + 64 
         -(4x³ + 28x²)
              -x² + 3x 
              -(-x² - 7x)
                     10x 

5. Bring Down and Repeat

  • Bring down the last term: Bring down the +64.
  • Repeat steps 2 and 3: Divide the new leading term (10x) by the leading term of the divisor (x).
  • Write the result above the line: 10x / x = 10, so write 10 above the line.
  • Multiply the divisor by 10: (x + 7) * 10 = 10x + 70.
  • Subtract the result:
             4x² - x + 10      
x + 7 | 4x³ + 27x² + 3x + 64 
         -(4x³ + 28x²)
              -x² + 3x 
              -(-x² - 7x)
                     10x + 64
                     -(10x + 70)
                            -6 

6. The Result

The final result of the division is:

4x² - x + 10 - 6/(x + 7)

Therefore, (4x³ + 27x² + 3x + 64) / (x + 7) = 4x² - x + 10 - 6/(x + 7)

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