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Use the polynomial identity x^2 + 2ax + a^2 = ( x + a )^2 to rewrite x^2 + 12x + 36 in its factored form

A. (X + 6)^2
B. (C + 4)^2
C. (X + 9)^2
D. (X + 12)^2

1 answer

To rewrite the polynomial \( x^2 + 12x + 36 \) in its factored form using the given identity \( x^2 + 2ax + a^2 = (x + a)^2 \), we can identify \( 2a \) and \( a^2 \) from the polynomial.

In the given polynomial:

  1. The coefficient of \( x \) is \( 12 \), so we set \( 2a = 12 \). Solving for \( a \) gives: \[ a = \frac{12}{2} = 6 \]

  2. The constant term is \( 36 \), which is \( a^2 \): \[ a^2 = 6^2 = 36 \]

Since both conditions match, we can rewrite the polynomial as: \[ x^2 + 12x + 36 = (x + 6)^2 \]

Thus, the correct factored form is:

A. \((X + 6)^2\)