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

Jasonbradley

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·

Jun 17, 2024

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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

5. 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 
   

6. 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 
   

7. 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 
   

8. 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).