Question

integrated ii > chapter 4: chapter 4 maintaining mathematical proficiency > section exercises 4 > exerc
factor $x^2 + 16x + 64$.
the factored form is \boxed{}.

Explanation:

Step1: Recall the perfect square trinomial formula

The formula for a perfect square trinomial is $a^2 + 2ab + b^2=(a + b)^2$.

Step2: Identify \(a\) and \(b\) in the given trinomial

For the trinomial \(x^2+16x + 64\), we have \(a = x\) (since \(a^2=x^2\)), and we need to find \(b\) such that \(2ab = 16x\) and \(b^2=64\).
From \(b^2 = 64\), we get \(b = 8\) (since \(8^2=64\)).
Checking \(2ab\): \(2\times x\times8=16x\), which matches the middle term.

Step3: Apply the perfect square trinomial formula

Using the formula \(a^2 + 2ab + b^2=(a + b)^2\) with \(a=x\) and \(b = 8\), we get \(x^2+16x + 64=(x + 8)^2\).

Answer:

$(x + 8)^2$