Dividing Polynomials: (7n^468n^3+46n^27n18)/(n9)
This article will guide you through the process of dividing the polynomial 7n^468n^3+46n^27n18 by n9.
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^468n^3+46n^27n18) by (n9)

Set up the division: Write the problem as a long division problem:
____________ n9  7n^4  68n^3 + 46n^2  7n  18

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 _______ n9  7n^4  68n^3 + 46n^2  7n  18

Multiply the quotient term by the divisor:
 Multiply the quotient term (7n^3) by the divisor (n9) to get 7n^4  63n^3.
 Write the result below the dividend.
7n^3 _______ n9  7n^4  68n^3 + 46n^2  7n  18 7n^4  63n^3 

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 _______ n9  7n^4  68n^3 + 46n^2  7n  18 7n^4  63n^3  5n^3 + 46n^2

Bring down the next term:
 Bring down the next term (7n) from the dividend.
7n^3 _______ n9  7n^4  68n^3 + 46n^2  7n  18 7n^4  63n^3  5n^3 + 46n^2  7n

Repeat steps 25:
 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 (n9) to get 5n^3 + 45n^2.
 Subtract the result from the previous step.
 Bring down the next term (18).
7n^3  5n^2 _______ n9  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

Continue repeating steps 25:
 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 (n9) to get n^2  9n.
 Subtract the result from the previous step.
7n^3  5n^2 + n _______ n9  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

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 (n9) to get 2n  18.
 Subtract the result from the previous step, resulting in a remainder of 0.
7n^3  5n^2 + n + 2 n9  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^468n^3+46n^27n18) divided by (n9) is 7n^3  5n^2 + n + 2. The remainder is 0.