(x^5+7)/(x^3-1) Long Division

6 min read Jun 17, 2024
(x^5+7)/(x^3-1) Long Division

Long Division of Polynomials: (x^5 + 7) / (x^3 - 1)

Long division of polynomials is a method used to divide one polynomial by another. This process can be used to simplify expressions, solve equations, and find factors of polynomials.

Let's work through the division of (x^5 + 7) / (x^3 - 1) step-by-step.

Setting up the Division

  1. Organize the polynomials: Arrange the terms of both polynomials in descending order of their exponents. If any terms are missing, fill in the gaps with a coefficient of 0.

         x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
       __________________________________
    x^3 - 1 |
    
  2. Start the division: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x^3). This gives us x^2. Write this term above the line, aligned with the x^5 term.

         x^2             
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
    
  3. Multiply the divisor by the term just written: Multiply (x^3 - 1) by x^2. This gives us x^5 - x^2. Write this result below the dividend, aligning the terms.

         x^2             
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
             x^5 - x^2          
    
  4. Subtract: Subtract the terms you just wrote from the dividend. Remember to change the signs of the terms you are subtracting.

         x^2             
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
             x^5 - x^2          
             -------
               0x^4 + 0x^3 + x^2 + 0x + 7 
    

Continue the Process

  1. Bring down the next term: Bring down the next term of the dividend (0x) to the result.

         x^2             
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
             x^5 - x^2          
             -------
               0x^4 + 0x^3 + x^2 + 0x + 7 
    
  2. Repeat the steps: Divide the leading term of the new dividend (0x^4) by the leading term of the divisor (x^3). This gives us 0. Write this term above the line, aligned with the x^4 term.

         x^2 + 0           
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
             x^5 - x^2          
             -------
               0x^4 + 0x^3 + x^2 + 0x + 7 
    
  3. Multiply and subtract: Multiply (x^3 - 1) by 0. Write this result below the new dividend, align the terms and subtract.

         x^2 + 0           
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
             x^5 - x^2          
             -------
               0x^4 + 0x^3 + x^2 + 0x + 7 
               0x^4 + 0x^3 + 0x  
               -------
                    x^2 + 0x + 7 
    
  4. Continue: Repeat the process until the degree of the remaining dividend is less than the degree of the divisor.

         x^2 + 0 + 1        
       __________________________________
    x^3 - 1 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7     
             x^5 - x^2          
             -------
               0x^4 + 0x^3 + x^2 + 0x + 7 
               0x^4 + 0x^3 + 0x  
               -------
                    x^2 + 0x + 7 
                    x^2 - 1       
                    -------
                           0x + 8   
    

Final Result

The final result of the long division is:

(x^5 + 7) / (x^3 - 1) = x^2 + 1 + (8 / (x^3 - 1))

The term (8 / (x^3 - 1)) is the remainder of the division. This means that the original polynomial can be expressed as the sum of the quotient (x^2 + 1) and the remainder (8 / (x^3 - 1)) multiplied by the divisor (x^3 - 1).