In Exercises 49–56, factor each perfect square trinomial. x^2+2x+1

Verified step by step guidance

1. Step 1: Identify the trinomial as a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In this case, $x^2+2x+1$ is a perfect square trinomial because it can be written as $(x+1)^2$.
2. Step 2: Identify the square root of the first term and the last term. The square root of $x^2$ is $x$ and the square root of $1$ is $1$.
3. Step 3: Check if the middle term is twice the product of the square roots of the first and last terms. In this case, $2x$ is indeed twice the product of $x$ and $1$.
4. Step 4: If the middle term is twice the product of the square roots of the first and last terms, then the trinomial can be factored into the square of a binomial. The binomial is formed by the square roots of the first and last terms, and the sign of the middle term. In this case, the binomial is $(x+1)$.
5. Step 5: Therefore, the factored form of the perfect square trinomial $x^2+2x+1$ is $(x+1)^2$.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It takes the form a^2 + 2ab + b^2, which factors to (a + b)^2. Recognizing this pattern is essential for factoring such expressions efficiently.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the case of perfect square trinomials, this involves identifying the binomial that, when squared, produces the trinomial.

Binomial

A binomial is a polynomial that consists of exactly two terms, typically in the form a + b or a - b. Understanding binomials is crucial for factoring, as perfect square trinomials can be rewritten as the square of a binomial, simplifying the expression significantly.