(x^2+5x-14)/(x-2) Long Division

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x² + 5x - 14) / (x - 2)

Long division with polynomials follows a similar process to long division with numbers. Let's divide (x² + 5x - 14) by (x - 2).

Steps:

1. Set up the division:

        _______
   x - 2 | x² + 5x - 14
   

2. Divide the leading terms:

   • The leading term of the divisor (x - 2) is 'x'.
   • The leading term of the dividend (x² + 5x - 14) is 'x²'.
   • x² / x = x, which goes above the line.

        x______
   x - 2 | x² + 5x - 14
   

3. Multiply the quotient (x) by the divisor (x - 2):

   • x * (x - 2) = x² - 2x. Write this result below the dividend.

        x______
   x - 2 | x² + 5x - 14
          x² - 2x 
   

4. Subtract:

   • Subtract the result (x² - 2x) from the dividend. Remember to change the signs when subtracting.

        x______
   x - 2 | x² + 5x - 14
          x² - 2x 
          -------
              7x - 14 
   

5. Bring down the next term:

   • Bring down the next term from the dividend (-14).

        x______
   x - 2 | x² + 5x - 14
          x² - 2x 
          -------
              7x - 14 
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (7x) by the leading term of the divisor (x). This gives us 7.
   • Multiply 7 by the divisor (x - 2) to get 7x - 14.
   • Subtract this result from the previous line.

        x + 7____
   x - 2 | x² + 5x - 14
          x² - 2x 
          -------
              7x - 14
              7x - 14
              -------
                  0 
   

7. The remainder is 0:

   • Since the remainder is 0, the division is complete.

Result:

Therefore, (x² + 5x - 14) / (x - 2) = x + 7.

This means that (x - 2) is a factor of (x² + 5x - 14).