Question

Use the Remainder Theorem to find the value of f(1/2) if f(x)=1/4x^3−1/8x+3.

Answer 1

To find the value of \( f\left(\frac{1}{2}\right) \) using the Remainder Theorem, we can evaluate the polynomial function \( f(x) = \frac{1}{4}x^3 - \frac{1}{8}x + 3 \) directly at \( x = \frac{1}{2} \).

Let's substitute \( x = \frac{1}{2} \) into the function:

\[
f\left(\frac{1}{2}\right) = \frac{1}{4}\left(\frac{1}{2}\right)^3 - \frac{1}{8}\left(\frac{1}{2}\right) + 3
\]

Calculating each part step by step:

1. Calculate \( \left(\frac{1}{2}\right)^3 \):
\[
\left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]

2. Substitute this value into the first term:
\[
\frac{1}{4}\left(\frac{1}{8}\right) = \frac{1}{32}
\]

3. Calculate the second term:
\[
-\frac{1}{8}\left(\frac{1}{2}\right) = -\frac{1}{16}
\]

Now, we can rewrite \( f\left(\frac{1}{2}\right) \):
\[
f\left(\frac{1}{2}\right) = \frac{1}{32} - \frac{1}{16} + 3
\]

To combine \( \frac{1}{32} \) and \( -\frac{1}{16} \), we need a common denominator:
- The common denominator of 32 and 16 is 32.

Convert \( -\frac{1}{16} \) to a fraction with a denominator of 32:
\[
-\frac{1}{16} = -\frac{2}{32}
\]

Now combine the fractions:
\[
f\left(\frac{1}{2}\right) = \frac{1}{32} - \frac{2}{32} + 3 = -\frac{1}{32} + 3
\]

Convert 3 into a fraction with a denominator of 32:
\[
3 = \frac{96}{32}
\]

Now combine:
\[
f\left(\frac{1}{2}\right) = -\frac{1}{32} + \frac{96}{32} = \frac{95}{32}
\]

Thus, the value of \( f\left(\frac{1}{2}\right) \) is:

\[
\boxed{\frac{95}{32}}
\]