91Ó°ÊÓ

Firstly, confirm that the divisor is of the form x - c. Here, it is \(x - 9\) so it is in the correct form. Write down the coefficients of the terms in the numerator and denominator. The coefficients for the numerator \(x^{3}-729\) can be written as [1, 0, 0, -729] since there are no x^2 and x terms. The number to be written in the division bracket, should be the solution to the equation \(x - 9 = 0\), which is 9.

Write 1 (from the coefficients) under the division bar. Multiply the number from the division bracket by this number and write the result under the next number in the coefficients. Add numbers in this column and repeat for the rest columns. Multiply once again and do the same with the remaining columns until you reach the end. In this case, you'll multiply 9 (from the division bracket) by 1 (under the division bar from previous step) and write the result under the 0. You get 9. Sum the column to get 9. Repeat these steps with the rest numbers. Multiply the 9 (current number under the division bar) by 9 (from the bracket) and get 81. Write it under the next 0. Sum the column to get 81. Then multiply 81 by 9 to get 729. Write it under the -729 and sum this column to get 0.

The result of the synthetic division is the quotient of the original division. The numbers written under the division bar make up the coefficients of the polynomial quotient. In this case, it's 1x^2 + 9x + 81. There is no remainder in this case, as the last column sums to 0. So the expression \(x^{3}-729\) divided by \(x-9\) equals \(x^{2} + 9x + 81\).