Final answer:
To evaluate the function f(x) = x^3 + 6x^2 + 12x + 25 using synthetic division, divide the coefficients of the function by the given value of x = -4.
Step-by-step explanation:
To evaluate the function f(x) using synthetic division, we will divide the coefficients of the function by the given value of x = -4.
First, let's set up the synthetic division table:
-4 | 1 6 12 25
|
Starting with the coefficient 1, we bring it down to the line:
-4 | 1 6 12 25
| 1
Next, we multiply the divisor -4 by the 1 and write the result below the next coefficient:
-4 | 1 6 12 25
| 1
|-4
Adding the two values, we get 2, which becomes the next coefficient:
-4 | 1 6 12 25
| 1 -4
|-4
----
1 2
Continuing this process, we repeat the steps until all coefficients have been evaluated:
-4 | 1 6 12 25
| 1 -4 -4
|-4 12
----
1 2 8
The last value, 8, is the remainder. The values in the bottom row represent the coefficients of the quotient polynomial. Therefore, the quotient polynomial is: x^2 + 2x + 8.