Use the Remainder Theorem to find the value of f(12)

To find the value of \( f(12) \) using the Remainder Theorem, we simply need to substitute \( x = 12 \) into the polynomial \( f(x) = 14x^3 - 18x + 3 \).

Let's calculate it step by step:

1. Calculate \( 14(12^3) \): \[ 12^3 = 1728 \quad \Rightarrow \quad 14 \times 1728 = 24192 \]

2. Calculate \( -18(12) \): \[ -18 \times 12 = -216 \]

3. Add the constant term \( 3 \): \[ f(12) = 24192 - 216 + 3 \] \[ f(12) = 24192 - 216 = 23976 \] \[ f(12) = 23976 + 3 = 23979 \]

Thus, \( f(12) = 23979 \).

However, this value is not in the options you provided—let's reevaluate the options you mentioned. The potential answers don't include \( 23979 \). Let's check if there might have been an error—please confirm if we used the correct polynomial.

If you are looking for one of the responses you listed, it seems that none match our calculation. If any of the steps appear mistaken, or there's a misunderstanding in the question context or polynomial, feel free to clarify!

To find the value of \( f(12) \) for the function \( f(x) = \frac{14}{x^3} - \frac{1}{8}x + 3 \), we substitute \( x = 12 \) into the function.

Let's calculate it step by step.

1. Calculate \( \frac{14}{12^3} \): \[ 12^3 = 1728 \quad \Rightarrow \quad \frac{14}{1728} = \frac{14}{1728} \approx 0.0081 \]

2. Calculate \( -\frac{1}{8} \times 12 \): \[ -\frac{1}{8} \times 12 = -\frac{12}{8} = -\frac{3}{2} = -1.5 \]

3. Now add the constant term \( 3 \): \[ f(12) = \frac{14}{1728} - 1.5 + 3 \] \[ f(12) = \frac{14}{1728} + 3 - 1.5 \] \[ f(12) = \frac{14}{1728} + 1.5 \] Converting 1.5 to a fraction: \[ 1.5 = \frac{3}{2} = \frac{3 \times 864}{2 \times 864} = \frac{2592}{1728} \]

4. Now add the fractions: \[ f(12) = \frac{14 + 2592}{1728} = \frac{2606}{1728} \]

5. This fraction can be simplified: \[ \frac{2606}{1728} = \frac{1303}{864} \quad \text{(dividing both numerator and denominator by 2)} \]

Thus, the value of \( f(12) \) is \( \frac{1303}{864} \).

Please check if this value aligns with your expected answers. If you have specific options to choose from, please provide them!