Use the polynomial identity x2+2ax+a2=(x+a)(x+a) to rewrite x2+4x+4. Which polynomial does this establish?(1 point)

Question

Use the polynomial identity x2+2ax+a2=(x+a)(x+a) to rewrite x2+4x+4. Which polynomial does this establish?(1 point)
Responses

x2+4x+4=(x+4)(x+4)
x squared plus 4 x plus 4 equals left parenthesis x plus 4 right parenthesis left parenthesis x plus 4 right parenthesis

x2+4x+4=(x−2)(x−2)
x squared plus 4 x plus 4 equals left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis

x2+4x+4=(x−4)(x−4)
x squared plus 4 x plus 4 equals left parenthesis x minus 4 right parenthesis left parenthesis x minus 4 right parenthesis

x2+4x+4=(x+2)(x+2)

GPT-4o mini · Answer

To rewrite the polynomial $ x^2 + 4x + 4 $ using the polynomial identity $ x^2 + 2ax + a^2 = (x + a)(x + a) $, we need to identify $ 2a $ and $ a^2 $ in the expression. In the given polynomial $ x^2 + 4x + 4 $: - The coefficient of $ x $ is $ 4 $. - The constant term is $ 4 $. We can identify $ 2a $ and $ a^2 $ as follows: - Since $ 2a = 4 $, we have $ a = 2 $. - Since $ a^2 = 4 $, we can also verify that $ 2^2 = 4 $. Applying the identity: $$ x^2 + 4x + 4 = x^2 + 2(2)x + 2^2 = (x + 2)(x + 2) $$ Thus, we can write: $$ x^2 + 4x + 4 = (x + 2)(x + 2) $$ Therefore, the correct response is: **$ x^2 + 4x + 4 = (x + 2)(x + 2) $**.