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

6 min read Jun 16, 2024
(7n^4-68n^3+46n^2-7n-18)/(n-9)

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.

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