Problem 77 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(x^{2}+2 x+1\)

Step-by-Step Solution

Answer

The expression \(x^2 + 2x + 1\) factors to \((x + 1)^2\).

Step 1: Identify the Quadratic Expression

The given expression is a quadratic expression in the form of a trinomial: \(x^2 + 2x + 1\). Our goal is to factor this expression into a product of two binomials.

Step 2: Recognize Patterns in Quadratics

Identify that the expression is a perfect square trinomial, which follows the pattern \((a + b)^2 = a^2 + 2ab + b^2\). In this problem, the expression \(x^2 + 2x + 1\) resembles this pattern.

Step 3: Determine \(a\) and \(b\)

For the expression \(x^2 + 2x + 1\), compare it to the pattern \(a^2 + 2ab + b^2\). Here, \(a = x\) and \(b = 1\), since \(a^2 = x^2\) and \(b^2 = 1\). The middle term, \(2ab\), is \(2(x)(1) = 2x\).

Step 4: Write as a Perfect Square

Using the values of \(a\) and \(b\), write the expression as a square of a binomial: \((x + 1)^2\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, usually in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are fundamental in algebra and appear frequently in various mathematical applications. Recognizing and solving quadratic equations is a key skill in algebra.

The standard form makes it easy to identify the quadratic term (\(ax^2\)), the linear term (\(bx\)), and the constant term (\(c\)). You can solve quadratic equations using several methods, such as factoring, completing the square, or using the quadratic formula. In our original exercise, we are dealing specifically with a quadratic trinomial, which can often be factored into binomial expressions as part of solving it.

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic trinomial that can be expressed as the square of a binomial expression. It follows the pattern \((a + b)^2 = a^2 + 2ab + b^2\). This pattern is key to recognizing the trinomial structure as one that can be perfectly factored.

For example, in our exercise, the expression \(x^2 + 2x + 1\) fits the perfect square trinomial pattern. Here, \(a = x\) and \(b = 1\), because \(x^2\) matches \(a^2\) and the term \(1\) matches \(b^2\). The middle term, \(2ab\), corresponds to \(2 \times x \times 1\), which equals \(2x\).

Recognizing perfect square trinomials is useful because they simplify neatly into the square of a binomial, making them easy to factor and solve. They often appear in problems where symmetry and simplification are required.

Binomial Expressions

Binomial expressions consist of two terms connected by either a plus or minus sign, such as \(x + 1\) or \(x - 2\). In the context of factoring quadratics, converting quadratic expressions into binomials is a common goal.

Once a quadratic expression like our trinomial is recognized as a perfect square, it can be rewritten as a binomial expression squared. For instance, \(x^2 + 2x + 1\) can be factored into \((x + 1)^2\). This indicates that the expression is the square of the binomial \(x + 1\).

Understanding binomial expressions is crucial for simplifying polynomials, solving equations, and in calculus, as they form the basis for factoring more complex algebraic expressions. They are building blocks for much more intricate mathematical ideas and applications.