(x^2+7x-5)/(x-2) Long Division

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x^2 + 7x - 5) / (x - 2)

Long division of polynomials is a method for dividing two polynomials. It is similar to the long division of numbers that you learned in elementary school.

In this article, we will demonstrate how to perform the long division of (x^2 + 7x - 5) by (x - 2).

Steps:

1. Set up the division: Write the dividend (x^2 + 7x - 5) inside the division symbol and the divisor (x - 2) outside.

        ___________
   x - 2 | x^2 + 7x - 5 
   

2. Divide the leading terms: Divide the leading term of the dividend (x^2) by the leading term of the divisor (x). This gives you x. Write this above the line.

        x________
   x - 2 | x^2 + 7x - 5 
   

3. Multiply the quotient by the divisor: Multiply the quotient (x) by the divisor (x - 2). This gives you x^2 - 2x. Write this below the dividend, aligning the terms.

        x________
   x - 2 | x^2 + 7x - 5 
           x^2 - 2x
   

4. Subtract: Subtract the result from the dividend. Remember to change the signs of the terms you are subtracting. This leaves you with 9x.

        x________
   x - 2 | x^2 + 7x - 5 
           x^2 - 2x
           -------
                  9x 
   

5. Bring down the next term: Bring down the next term of the dividend (-5).

        x________
   x - 2 | x^2 + 7x - 5 
           x^2 - 2x
           -------
                  9x - 5
   

6. Repeat steps 2-5: Divide the leading term of the new dividend (9x) by the leading term of the divisor (x). This gives you 9. Write this above the line.

        x + 9_____
   x - 2 | x^2 + 7x - 5 
           x^2 - 2x
           -------
                  9x - 5
                  9x - 18
   

   Multiply the quotient (9) by the divisor (x - 2). This gives you 9x - 18. Write this below the new dividend, aligning the terms.

        x + 9_____
   x - 2 | x^2 + 7x - 5 
           x^2 - 2x
           -------
                  9x - 5
                  9x - 18
   

   Subtract the result from the new dividend. This leaves you with 13.

        x + 9_____
   x - 2 | x^2 + 7x - 5 
           x^2 - 2x
           -------
                  9x - 5
                  9x - 18
                  -------
                         13
   

7. The result: The final result is the quotient (x + 9) plus the remainder (13) divided by the divisor (x - 2).

   (x^2 + 7x - 5) / (x - 2) = x + 9 + 13/(x - 2)

Conclusion:

By following the steps of long division, we have successfully divided (x^2 + 7x - 5) by (x - 2). The result is x + 9 + 13/(x - 2). This method is a valuable tool for simplifying and manipulating polynomial expressions.