(3x2−14x−5)÷(x−5)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (3x²−14x−5) ÷ (x−5)

This article will guide you through the process of dividing the polynomial (3x²−14x−5) by (x−5) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but instead of dividing numbers, we're dividing polynomials.

Step-by-Step Solution

1. Set up the division:

       __________
   x-5 | 3x² - 14x - 5 
   

2. Divide the leading terms:

   • The leading term of the divisor (x-5) is 'x'.
   • The leading term of the dividend (3x² - 14x - 5) is '3x²'.
   • Divide '3x²' by 'x' to get '3x'.

       3x ______
   x-5 | 3x² - 14x - 5 
   

3. Multiply the quotient by the divisor:

   • Multiply '3x' (the quotient) by (x-5) to get '3x² - 15x'.

       3x ______
   x-5 | 3x² - 14x - 5 
          3x² - 15x 
   

4. Subtract:

   • Subtract '3x² - 15x' from the dividend.

       3x ______
   x-5 | 3x² - 14x - 5 
          3x² - 15x 
          -------
              x - 5 
   

5. Bring down the next term:

   • Bring down the '-5' from the dividend.

       3x ______
   x-5 | 3x² - 14x - 5 
          3x² - 15x 
          -------
              x - 5 
   

6. Repeat steps 2-5:

   • The leading term of the new dividend is 'x'.
   • Divide 'x' by 'x' to get '1'.

       3x + 1 ______
   x-5 | 3x² - 14x - 5 
          3x² - 15x 
          -------
              x - 5 
              x - 5
   

   • Multiply '1' by (x-5) to get 'x-5'.

       3x + 1 ______
   x-5 | 3x² - 14x - 5 
          3x² - 15x 
          -------
              x - 5 
              x - 5
   

   • Subtract 'x-5' from the dividend.

       3x + 1 ______
   x-5 | 3x² - 14x - 5 
          3x² - 15x 
          -------
              x - 5 
              x - 5
              ----
                0 
   

7. The Result:

   • Since the remainder is 0, the division is complete.
   • Therefore, (3x²−14x−5) ÷ (x−5) = 3x + 1.

Conclusion

Using polynomial long division, we found that (3x²−14x−5) divided by (x−5) equals 3x + 1. This method allows us to divide polynomials and express the result as a quotient and a remainder.