Use synthetic division f(x)=x^3+15x^2+71x+105 by x+7
To use synthetic division, we first need to set up the synthetic division table by writing down the coefficients of the polynomial in descending order:
7 | 1 15 71 105
Now, we perform the synthetic division operation following these steps:
Step 1: Bring down the first coefficient (1) to start the process:
7 | 1 15 71 105
1
Step 2: Multiply the divisor (7) by the value just brought down (1), and write the result below the next coefficient:
7 | 1 15 71 105
-7
---
8
Step 3: Add the result from step 2 to the next coefficient (15), and write the sum below the line:
7 | 1 15 71 105
-7
---
8 8
Step 4: Repeat steps 2 and 3 for the next coefficient (8):
7 | 1 15 71 105
-7 8
---
8 8 1
Step 5: Repeat steps 2 and 3 for the last coefficient (1):
7 | 1 15 71 105
-7 8 1
---
8 8 1 112
The final row represents the coefficients of the quotient polynomial. Therefore, the result of dividing f(x) = x^3 + 15x^2 + 71x + 105 by x + 7 is:
quotient = x^2 + 8x + 1 and remainder = 112