Problem 50 - College Algebra Essentials

Question

Factor each perfect square trinomial. $$x^{2}+4 x+4$$

Step-by-Step Solution

Answer

The factorized form of the given perfect square trinomial \(x^{2}+4 x+4\) is \((x+2)^2\).

Step 1: Recognize the structure

Recognize that the trinomial follows the structure of a perfect square trinomial, i.e., in the form of \((a+b)^2\). Here, \(a\) can be 'x', and \(b\) can be '2', as the square of 2 gives the third term '4'.

Step 2: Calculate 'b'

The coefficient of the middle term, '4', is twice the product of \(a\) and \(b\). That is, \(2ab = 4\), so \(b\) is '2'.

Step 3: Write in the factored form

Using the structure of the perfect square trinomial, which is \((a+b)^2\), substitute \(a\) and \(b\) with 'x' and '2' to get the factored form. Thus, the factored form is \( (x+2)^2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is akin to breaking down numbers into their prime factors, except the 'numbers' are algebraic expressions. When factoring polynomials, we aim to express the polynomial as a product of its irreducible factors. It's critical to recognize patterns to make this process easier.

One familiar pattern includes the perfect square trinomial. It emerges when a binomial is squared resulting in a trinomial of the form \(a^2 + 2ab + b^2\). The key to factoring this type of expression lies in identifying the square roots of the first and last term and ensuring that the middle term is twice the product of these square roots.

For example, in our case, the polynomial \(x^2 + 4x + 4\) fits this pattern. We identify \(a = x\) and \(b = 2\) because \(x^2\) is the square of \(x\), and \(4\) is the square of \(2\). Furthermore, the middle term \(4x\) is indeed twice the product of \(x\) and \(2\). From this, it can be concluded that the trinomial is a perfect square and can be factored as \( (x + 2)^2\).

Algebraic Expressions

Algebraic expressions form the bedrock of algebra and include numbers, variables, and arithmetic operations. They can be as simple as \(x + 1\) or as complex as the expression for the area of a circle. Understanding how to manipulate these expressions is crucial for solving equations and understanding how algebraic operations work.

An important aspect is combining like terms—terms that have the same variable raised to the same power. Additionally, recognizing certain patterns in expressions can simplify the solving process. Our current manifestation of an algebraic expression, the perfect square trinomial, is a specialized case that benefits from pattern recognition.

As we dealt with the perfect square \(x^2 + 4x + 4\), this wasn’t just a random arrangement of terms. It was carefully constructed from the binomial \(x+2\) and indicates a square of that binomial, showcasing the interconnectedness within algebraic expressions.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The solutions to these equations can be found via various methods including factoring, completing the square, or using the quadratic formula.

Factoring is one of the most direct methods when applicable, especially when the quadratic is a perfect square trinomial. To factor such an expression is to write it in a form that can easily be set to zero and solved.

In the exercise \(x^2 + 4x + 4 = 0\), factoring the trinomial as \( (x + 2)^2\) simplifies solving the equation to finding a number that when added to 2 and squared results in zero. The solution is then straightforward: \(x + 2 = 0\), which gives us \(x = -2\). This straightforward process exemplifies why recognizing perfect square trinomials in quadratic equations is a valuable skill.