(x^3-x^2-5x-3) Divided By (x^2+2x+1)

5 min read Jun 17, 2024
(x^3-x^2-5x-3) Divided By (x^2+2x+1)

Dividing Polynomials: (x^3-x^2-5x-3) ÷ (x^2+2x+1)

This article will guide you through the process of dividing the polynomial (x^3-x^2-5x-3) by (x^2+2x+1). We'll use the method of long division to achieve this.

1. Setting Up the Long Division

Start by writing the division problem in the standard long division format:

            _______
x^2+2x+1 | x^3-x^2-5x-3 

2. Dividing the Leading Terms

  • Focus on the leading terms of both the divisor and the dividend: x^2 (from the divisor) and x^3 (from the dividend).
  • Divide the leading term of the dividend by the leading term of the divisor: x^3 ÷ x^2 = x.
  • Write this quotient (x) above the x^2 term in the dividend:
            x      
x^2+2x+1 | x^3-x^2-5x-3 

3. Multiply and Subtract

  • Multiply the quotient (x) by the entire divisor (x^2+2x+1): x * (x^2+2x+1) = x^3 + 2x^2 + x.
  • Write the result below the dividend, aligning terms with their corresponding degrees:
            x      
x^2+2x+1 | x^3-x^2-5x-3 
            x^3+2x^2+x
  • Subtract the entire expression from the dividend: (x^3 - x^2 - 5x - 3) - (x^3 + 2x^2 + x) = -3x^2 - 6x - 3.
            x      
x^2+2x+1 | x^3-x^2-5x-3 
            x^3+2x^2+x
            -----------------
                  -3x^2-6x-3

4. Bring Down the Next Term

  • Bring down the next term from the dividend (-3) to the bottom row:
            x      
x^2+2x+1 | x^3-x^2-5x-3 
            x^3+2x^2+x
            -----------------
                  -3x^2-6x-3 

5. Repeat Steps 2-4

  • Focus on the leading term of the new dividend (-3x^2) and the leading term of the divisor (x^2): -3x^2 ÷ x^2 = -3.
  • Write this quotient (-3) next to the x in the quotient:
            x - 3    
x^2+2x+1 | x^3-x^2-5x-3 
            x^3+2x^2+x
            -----------------
                  -3x^2-6x-3 
  • Multiply the new quotient (-3) by the entire divisor: -3 * (x^2+2x+1) = -3x^2 - 6x - 3.
  • Write the result below the new dividend:
            x - 3    
x^2+2x+1 | x^3-x^2-5x-3 
            x^3+2x^2+x
            -----------------
                  -3x^2-6x-3
                  -3x^2-6x-3
  • Subtract the entire expression: (-3x^2 - 6x - 3) - (-3x^2 - 6x - 3) = 0.
            x - 3    
x^2+2x+1 | x^3-x^2-5x-3 
            x^3+2x^2+x
            -----------------
                  -3x^2-6x-3
                  -3x^2-6x-3
                  ----------
                        0

6. The Result

We've reached a remainder of 0, indicating that the division is complete. Therefore, we can conclude:

** (x^3 - x^2 - 5x - 3) ÷ (x^2 + 2x + 1) = x - 3**

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