Find the Synthetic Division:
(10x^4-50x^3-800)/(x-6)
To perform synthetic division, follow these steps:
Step 1: Write down the coefficients of the polynomial in descending order, including any missing terms. In this case, the polynomial is 10x^4 - 50x^3 - 800. So, the coefficients are: 10, -50, 0, 0, -800.
Step 2: Determine the divisor, which is x - 6. Set it equal to zero and solve for x: x - 6 = 0. By adding 6 to both sides, you get x = 6. This means x - 6 is the divisor.
Step 3: Set up the table for synthetic division. Write down the value of the divisor, 6, on the left side, and then write down the coefficients on the right side, starting with the highest power of x.
6 | 10 -50 0 0 -800
Step 4: Bring down the first coefficient, which is 10. Write it below the division bar.
6 | 10 -50 0 0 -800
|___
Step 5: Multiply the divisor, 6, with the number below the division bar (10). Write the result (60) above the second coefficient, which is -50. Then, add these two numbers: -50 + 60 = 10. Write the sum below the dashed line.
6 | 10 -50 0 0 -800
|___ 60
10
Step 6: Repeat Step 5 for each coefficient, bringing down the next coefficient and performing the multiplication and addition.
6 | 10 -50 0 0 -800
|___ 60 600
10
6 | 10 -50 0 0 -800
|___ 60 600 3600
10
6 | 10 -50 0 0 -800
|___ 60 600 3600 21600
10
6 | 10 -50 0 0 -800
|___ 60 600 3600 21600 124800
Step 7: The last number, 124800, represents the remainder of the division.
Therefore, the result of the synthetic division is:
10x^3 + 60x^2 + 600x + 3600 + 21600/(x - 6) with a remainder of 124800.