Problem 49 - College Algebra Essentials

Question

Factor each perfect square trinomial. $$x^{2}+2 x+1$$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(x^{2}+2 x+1\) is \((x + 1)^{2}\).

Step 1: Identify the terms

A perfect square trinomial is a trinomial of the form \(a^{2}+2ab+b^{2}\). For the trinomial \(x^{2}+2 x+1\), the corresponding values are \(a = x\) (as it's square is \(x^{2}\)), \(b = 1\) (as it's square is \(1\)) and \(2ab = 2 x \times 1 = 2x\). So \(x^{2}+2 x+1\) is indeed a perfect square trinomial.

Step 2: Factor the trinomial

A perfect square trinomial \((a^{2}+2ab+b^{2})\) can be factored as \((a+b)^{2}\). Substitute \(a = x\) and \(b = 1\) into this formula to get: \((x + 1)^{2}\).

Step 3: Final Answer

So, the factored form of the perfect square trinomial \(x^{2}+2 x+1\) is \((x + 1)^{2}\).