Question

What value of c makes the polynomial below a perfect square?

x2 + 16x + c

Answer 1

To determine the value of \( c \) that makes the polynomial \( x^2 + 16x + c \) a perfect square, we can express a perfect square polynomial in the form \( (x + k)^2 \), where \( k \) is a constant.

Expanding \( (x + k)^2 \) gives us:

\[
(x + k)^2 = x^2 + 2kx + k^2
\]

We need our polynomial \( x^2 + 16x + c \) to match this format:

\[
x^2 + 2kx + k^2
\]

From comparing coefficients, we can see that:

1. The coefficient of \( x \) gives us the equation \( 2k = 16 \).
2. The constant term gives us \( k^2 = c \).

Let's solve for \( k \):

\[
2k = 16 \implies k = \frac{16}{2} = 8
\]

Now, we can find \( c \) using \( k^2 \):

\[
c = k^2 = 8^2 = 64
\]

Thus, the value of \( c \) that makes the polynomial \( x^2 + 16x + c \) a perfect square is

\[
\boxed{64}
\]