Problem 98 - Algebra and Trigonometry

Question

Factor and simplify each algebraic expression. $$\left(x^{2}+4\right)^{\frac{1}{2}}+\left(x^{2}+4\right)^{\frac{7}{2}}$$

Step-by-Step Solution

Answer

\(\left(x^{2}+4\right)^{\frac{1}{2}}[1+\left(x^{2}+4\right)^{3}]\)

Step 1: Identify the Common Factor

Look for a common term in each part of the expression. In the expression \(\left(x^{2}+4\right)^{\frac{1}{2}}+\left(x^{2}+4\right)^{\frac{7}{2}}\), the common part is \(\left(x^{2}+4\right)^{\frac{1}{2}}\).

Step 2: Factor out the common term

Factor out the common term from the expression by rewriting each term divided by the common factor plus the common factor itself. It results in: \(\left(x^{2}+4\right)^{\frac{1}{2}}[1+\left(x^{2}+4\right)^{3}]\).

Step 3: Simplifying the Factor

Apply the binomial theorem to simplify \((x^{2}+4)^{3}\). This theorem states that the power of a binomial can be expanded using the coefficients of the binomial theorem. It is easier to keep the factor as is, as the exercise only requires factoring not simplifying.

Key Concepts

Understanding and Identifying Common Factors

In algebra, factoring is a critical skill that simplifies expressions by identifying common components. A **common factor** is a term that appears in each part of an expression. Recognizing these factors is essential for simplifying complex equations.

In the given expression \(\left(x^{2}+4\right)^{\frac{1}{2}}+\left(x^{2}+4\right)^{\frac{7}{2}}\), the common factor is \(\left(x^{2}+4\right)^{\frac{1}{2}}\). This is because this term is a part of both components of the expression.

By factoring out \(\left(x^{2}+4\right)^{\frac{1}{2}}\), the expression is simplified to \(\left(x^{2}+4\right)^{\frac{1}{2}}[1+\left(x^{2}+4\right)^{3}]\). This process makes it easier to manipulate or evaluate the expression in further steps of solving.

The Binomial Theorem Explained

The **binomial theorem** is a fundamental piece of algebra that provides a way to expand expressions that are raised to a power. It details how powers of a binomial can be expressed as a series of terms.

For example, the binomial theorem allows you to expand a term like \((x^{2}+4)^{3}\) into a sum of polynomial terms.

Each term in the expansion involves coefficients that come from Pascal's Triangle or can be calculated using factorial functions. Although this expansion wasn't needed for the current exercise, understanding this theorem is beneficial for complex algebraic manipulations.

Using the binomial theorem not only simplifies calculations but also aids in understanding patterns within polynomial expansions and how different algebraic entities combine together.

Algebraic Simplification Techniques

The process of **algebraic simplification** involves reducing expressions to their most compact form while preserving equivalence. It typically includes processes like factoring, expanding, combining like terms, and cancelling terms when permissible.

Simplifying an expression makes it easier to work with, understand, and solve if it is part of a broader equation or system.

In the solved exercise, the simplification involved using the common factor \(\left(x^{2}+4\right)^{\frac{1}{2}}\) to express the original expression in a streamlined manner: \(\left(x^{2}+4\right)^{\frac{1}{2}}[1+\left(x^{2}+4\right)^{3}]\).

To further simplify or solve similar expressions, one might employ additional techniques such as evaluating numeric values, rearranging terms, or applying mathematical properties like distribution or commutativity to transform it into the simplest possible form.