(7n^4-68n^3+46n^2-7n-18)/(n-9)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (7n^4-68n^3+46n^2-7n-18)/(n-9)

This article will guide you through the process of dividing the polynomial 7n^4-68n^3+46n^2-7n-18 by n-9.

Understanding Polynomial Division

Polynomial division is a method for dividing a polynomial by another polynomial of a lower or equal degree. This process is similar to long division with numbers.

Steps to Divide (7n^4-68n^3+46n^2-7n-18) by (n-9)

1. Set up the division: Write the problem as a long division problem:

        ____________
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
   

2. Divide the leading terms:

   • Divide the leading term of the dividend (7n^4) by the leading term of the divisor (n). This gives us 7n^3.
   • Write 7n^3 above the dividend.

        7n^3 _______
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
   

3. Multiply the quotient term by the divisor:

   • Multiply the quotient term (7n^3) by the divisor (n-9) to get 7n^4 - 63n^3.
   • Write the result below the dividend.

        7n^3 _______
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
         7n^4 - 63n^3
         ---------
   

4. Subtract:

   • Subtract the result from the previous step from the dividend.
   • Change the signs of the terms in the bottom row and add.

        7n^3 _______
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
         7n^4 - 63n^3
         ---------
              -5n^3 + 46n^2 
   

5. Bring down the next term:

   • Bring down the next term (-7n) from the dividend.

        7n^3 _______
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
         7n^4 - 63n^3
         ---------
              -5n^3 + 46n^2 - 7n 
   

6. Repeat steps 2-5:

   • Divide the new leading term (-5n^3) by the leading term of the divisor (n) to get -5n^2.
   • Multiply the new quotient term (-5n^2) by the divisor (n-9) to get -5n^3 + 45n^2.
   • Subtract the result from the previous step.
   • Bring down the next term (-18).

        7n^3 - 5n^2 _______
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
         7n^4 - 63n^3
         ---------
              -5n^3 + 46n^2 - 7n 
              -5n^3 + 45n^2
              ---------
                       n^2 - 7n - 18
   

7. Continue repeating steps 2-5:

   • Divide the new leading term (n^2) by the leading term of the divisor (n) to get n.
   • Multiply the new quotient term (n) by the divisor (n-9) to get n^2 - 9n.
   • Subtract the result from the previous step.

        7n^3 - 5n^2 + n _______
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
         7n^4 - 63n^3
         ---------
              -5n^3 + 46n^2 - 7n 
              -5n^3 + 45n^2
              ---------
                       n^2 - 7n - 18
                       n^2 - 9n
                       ---------
                              2n - 18
   

8. Final step:

   • Divide the new leading term (2n) by the leading term of the divisor (n) to get 2.
   • Multiply the new quotient term (2) by the divisor (n-9) to get 2n - 18.
   • Subtract the result from the previous step, resulting in a remainder of 0.

        7n^3 - 5n^2 + n + 2 
   n-9 | 7n^4 - 68n^3 + 46n^2 - 7n - 18 
         7n^4 - 63n^3
         ---------
              -5n^3 + 46n^2 - 7n 
              -5n^3 + 45n^2
              ---------
                       n^2 - 7n - 18
                       n^2 - 9n
                       ---------
                              2n - 18
                              2n - 18
                              ---------
                                  0
   

Conclusion

Therefore, the quotient of (7n^4-68n^3+46n^2-7n-18) divided by (n-9) is 7n^3 - 5n^2 + n + 2. The remainder is 0.