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

Verified step by step guidance

1. Recognize that the given expression \(x^2 + 2x + 1\) is a perfect square trinomial. A perfect square trinomial takes the form \(a^2 + 2ab + b^2\) and factors as \((a + b)^2\).
2. Identify the terms in the trinomial: \(x^2\) is the square of \(x\), and \(1\) is the square of \(1\).
3. Check the middle term \$2x\( to see if it matches \)2ab\(, where \)a = x\( and \)b = 1$. Since \(2 \times x \times 1 = 2x\), it fits the pattern.
4. Write the factored form using the pattern \((a + b)^2\), substituting \(a = x\) and \(b = 1\) to get \((x + 1)^2\).
5. Verify by expanding \((x + 1)^2\) to ensure it equals the original trinomial \(x^2 + 2x + 1\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form a² + 2ab + b², which factors into (a + b)². Recognizing this pattern helps simplify factoring problems quickly.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting them as a product of two binomials. For perfect square trinomials, this process is straightforward because the trinomial matches a specific pattern, allowing direct factoring without trial and error.

Identifying Coefficients and Terms

To factor correctly, it is essential to identify the coefficients of the quadratic and linear terms and the constant term. This helps determine if the trinomial fits the perfect square pattern by checking if the middle term equals twice the product of the square roots of the first and last terms.