Question

guided example: factor each expression.
a) ( x^2 - 36 )
b) ( x^2 + 36 )
3 ( x^2 + 18x + 81 )

Explanation:

Step1: Identify the form

The expression is \(x^2 + 18x + 81\), which looks like a perfect square trinomial. The form of a perfect square trinomial is \(a^2 + 2ab + b^2=(a + b)^2\).

Step2: Match the terms

Here, \(a = x\) (since \(x^2=(x)^2\)), and \(2ab = 18x\). Substituting \(a = x\) into \(2ab = 18x\), we get \(2\times x\times b = 18x\). Dividing both sides by \(2x\) (assuming \(x
eq0\)), we find \(b = 9\). And \(b^2=9^2 = 81\), which matches the constant term.

Step3: Factor the expression

Using the perfect square trinomial formula, \(x^2 + 18x + 81=(x + 9)^2\).

Answer:

\((x + 9)^2\)