Problem 27 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\((x^2 + 2)^2\) is the factorization.

Step 1: Recognize the Form

The trinomial is in the form of a perfect square trinomial. Let's express it in the form \(x^4 + 4x^2 + 4 = (x^2)^2 + 4x^2 + 2^2.\)This suggests a possible factorization as a square.

Step 2: Apply the Perfect Square Formula

Recall the perfect square formula:\((a + b)^2 = a^2 + 2ab + b^2.\)Here, we identify \(a = x^2\) and \(b = 2\). Check:\(2ab = 2(x^2)(2) = 4x^2,\)which matches our trinomial's middle term. Therefore, it can be factored as:\((x^2 + 2)^2.\)

Step 3: Analyze the Factorization

We found that the trinomial can be written as:\((x^2 + 2)^2,\)which means the trinomial is a perfect square. Therefore, the complete factorization of the trinomial is \((x^2 + 2)(x^2 + 2).\)

Key Concepts

Perfect Square Trinomial

A **perfect square trinomial** is a special form of polynomial that takes the form \(a^2 + 2ab + b^2\). It can also be written as \((a + b)^2\). Identifying a perfect square trinomial makes factoring much easier, as it involves recognizing the pattern and applying the formula directly to convert it into a squared expression.

In the given exercise, the trinomial \(x^4 + 4x^2 + 4\) is structured to resemble this pattern. Notice how it fits the perfect square form with \(a = x^2\) and \(b = 2\). These values satisfy the middle term's condition of \(2ab = 4x^2\), allowing us to quickly factor the expression into \((x^2 + 2)^2\).

Having a firm grasp on recognizing perfect square trinomials simplifies many algebraic manipulations needed when working with polynomials.

Algebraic Expressions

**Algebraic expressions** are combinations of variables, constants, and operations such as addition or multiplication. They form the basis of algebra and are found everywhere in mathematics. Understanding algebraic expressions is crucial for working with equations and polynomials.

In the context of factoring, algebraic expressions must sometimes be rewritten to identify patterns or forms. For example, the expression \(x^4 + 4x^2 + 4\) needs to be recognized as a perfect square trinomial. These expressions can be complex combinations, but breaking them down into smaller, recognizable components can make the task much more manageable. This is where knowledge of patterns like the perfect square trinomial plays a significant role.

It is important to familiarize yourself with different forms and the properties of algebraic expressions to deal with varying levels of complexity effectively.

Polynomial Factorization

**Polynomial factorization** refers to the process of breaking down a polynomial into a product of simpler polynomials. The goal is to express the polynomial as a multiplication of factors, which are easier to analyze or solve for their roots.

For instance, the trinomial \(x^4 + 4x^2 + 4\) in the exercise can be factored as \((x^2 + 2)(x^2 + 2)\), simplifying the original polynomial into a product of identical factors. Factorization often involves recognizing certain patterns or structures within the polynomial, such as difference of squares, perfect square trinomials, and others, to decompose them effectively.

Mastering polynomial factorization techniques is crucial for solving higher-degree polynomial equations, and for simplifying expressions in calculus and other advanced mathematical fields. The ability to factor accurately and efficiently opens doors to solutions that might otherwise remain hidden.