Problem 49 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The factorized form of \(x^{2}+2 x+1\) is \((x+1)^2\).

Step 1: Identify the three terms of the trinomial

In the trinomial \(x^{2}+2 x+1\), the three terms are \(x^{2}\), \(2x\), and \(1\).

Step 2: Check if the expression is a perfect square trinomial

A perfect square trinomial follows the form \(a^{2} + 2ab + b^{2}\). Here, \(a\) is \(x\) since \(x^{2}\) is the square of \(x\). Then \(b\) should be \(1\) because \(1\) is the square of \(1\), and the middle term \(2x\) equals to \(2 * a * b = 2*x*1\), which proves this trinomial is a perfect square.

Step 3: Apply the formula for factoring perfect square trinomials

The trinomial \(a^{2} + 2ab + b^{2}\) can be factored into \((a + b)^{2}\). Here, we substitute \(a = x\) and \(b = 1\), the expression becomes: \((x+1)^2\).