(x2+x−17)÷(x−4)

Jasonbradley

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Jun 17, 2024

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Polynomial Long Division: (x^2 + x - 17) ÷ (x - 4)

This article will guide you through the process of dividing the polynomial (x^2 + x - 17) by (x - 4) using long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to the long division you learned in elementary school for numbers. The goal is to find the quotient and remainder of the division.

Step-by-Step Solution:

1. Set up the division:

        _________
   x - 4 | x^2 + x - 17
   

2. Divide the leading terms:

   • x (from the divisor) goes into x^2 (from the dividend) x times.
   • Write x above the x^2 term in the quotient.

        x       
   x - 4 | x^2 + x - 17
   

3. Multiply the quotient term by the divisor:

   • x * (x - 4) = x^2 - 4x

        x       
   x - 4 | x^2 + x - 17
          x^2 - 4x
   

4. Subtract the result from the dividend:

   • (x^2 + x) - (x^2 - 4x) = 5x

        x       
   x - 4 | x^2 + x - 17
          x^2 - 4x
          -------
                 5x
   

5. Bring down the next term:

   • Bring down the -17 from the dividend.

        x       
   x - 4 | x^2 + x - 17
          x^2 - 4x
          -------
                 5x - 17
   

6. Repeat the process:

   • x (from the divisor) goes into 5x (from the dividend) 5 times.
   • Write 5 next to the x in the quotient.

        x + 5
   x - 4 | x^2 + x - 17
          x^2 - 4x
          -------
                 5x - 17
                 5x - 20 
   

7. Multiply and subtract:

   • 5 * (x - 4) = 5x - 20
   • (5x - 17) - (5x - 20) = 3

        x + 5
   x - 4 | x^2 + x - 17
          x^2 - 4x
          -------
                 5x - 17
                 5x - 20 
                 -------
                      3 
   

8. The remainder:

   • We are left with a remainder of 3.

Result:

The division of (x^2 + x - 17) by (x - 4) yields a quotient of (x + 5) and a remainder of 3. This can be expressed as:

(x^2 + x - 17) ÷ (x - 4) = x + 5 + 3/(x - 4)

Conclusion:

Polynomial long division is a systematic method for dividing polynomials. By following the steps outlined above, you can successfully find the quotient and remainder of any polynomial division problem.