Sure, let's use synthetic division to solve the equation.
Firstly, we write down the coefficients of the cubic polynomial in the left column and the divisor in the frame (or box). The polynomial we want to divide is x³ - 7x² + 0*x + 41 and the divisor is x-6. The coefficients of the polynomial are 1 (from x³), -7 (from -7x²), 0 (since there is no x term) and 41 (the constant).
The divisor is x-6, so we put 6 into the frame.
Our synthetic division table looks like this:
1 -7 0 41
6 __ __
For synthetic division
1. Draw a line under the first coefficient (which is 1), and write it below the line. This is the first term of our answer.
2. Multiply the term we just wrote (which is also 1) by the divisor (which is 6) and write this result under the second original coefficient (which is -7). Write the result (-6) in the second row of the table.
3. Now add the second column (-7 and -6), the result is -13. Write this below the line. This is the second term of our quotient.
4. Repeat steps 2 and 3 for the entire table. After calculating, our table looks like this:
1 -7 0 41
6 __/ \__/ -78
________________
1 -13 -78 7
As a result, our quotient is 0.16667x² -1.16667x + 0 + 6.83333, or 0.16667x² -1.16667x + 6.83333.
The remainder is the only term remaining in the last column of the synthetic division table, 7. We divide it by the original divisor, 6, and find that the remainder is 0.
So, as a conclusion, when we divide x³ -7x² + 41 by x-6, we get a quotient of 0.16667x² -1.16667x + 6.83333 and a remainder of 0.