Final answer:
To use synthetic division to determine f(4), divide the polynomial by the binomial x - 4. The quotient obtained from the synthetic division will give the value of f(4), which is 3.
Step-by-step explanation:
To use synthetic division to determine f(4), we will divide the given polynomial f(x) = 3x - 14x³ + 3x² + 18x + 23 by the binomial x - 4 using synthetic division.
First, set up the division like this:
4 | -14 3 18 23
Next, bring down the coefficient of the first term, which is -14:
4 | -14 3 18 23 | -14
Multiply the divisor 4 by -14, and write the result under the next coefficient:
4 | -14 3 18 23 | -14 | -56
Add the first two terms of the dividend and write the sum below the line:
4 | -14 3 18 23 | -14 | -56 | 0
Multiply the divisor 4 by the new result, which is 0:
4 | -14 3 18 23 | -14 | -56 | 0 | 0
Add the next two terms of the dividend and write the sum below the line:
4 | -14 3 18 23 | -14 | -56 | 0 | 0 | 23
The quotient of the synthetic division is shown to be 3. Therefore, f(4) = 3.