To complete the square for the polynomial \( f^2 + 16f \), we first need to identify the coefficient of the \( f \) term.
- The coefficient of \( f \) is 16.
- To complete the square, we take half of this coefficient, which is \( \frac{16}{2} = 8 \).
- Next, we square this value: \( 8^2 = 64 \).
Now, we can add and subtract this value (64) to complete the square:
\[ f^2 + 16f = (f^2 + 16f + 64 - 64) = (f + 8)^2 - 64 \]
Thus, the value to fill in the blank to make the polynomial a perfect square is \( 64 \).
So the completed polynomial is: \[ f^2 + 16f + 64. \]
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