Problem 80 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \(\left(c+\frac{1}{c}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(c^2 + 2 + \frac{1}{c^2}\).

Step 1: Expand the Square

We begin with the expression \( \left(c + \frac{1}{c}\right)^2 \). This can be expanded using the identity \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = c\) and \(b = \frac{1}{c}\).

Step 2: Apply the Formula

Using the formula from the previous step, we substitute \(a\) and \(b\) to get: \[a^2 + 2ab + b^2 = c^2 + 2 \cdot c \cdot \frac{1}{c} + \left(\frac{1}{c}\right)^2.\] This simplifies to: \[c^2 + 2 + \frac{1}{c^2}.\]

Step 3: Simplify the Expression

Combine all the terms into a single expression to complete the simplification: \[c^2 + 2 + \frac{1}{c^2}.\] Here, we observed that the middle term 2 comes from the \(2 \times c \times \frac{1}{c}\) simplifying to 2.

Key Concepts

Expanding Binomials

The process of expanding binomials is crucial in algebra as it helps us express the square of a binomial expression as a trinomial. This typically involves using a well-known algebraic identity that relates to multiplying out the terms of a binomial raised to a power, usually 2. The key identity used is \((a + b)^2 = a^2 + 2ab + b^2\). This formula tells us exactly how to expand the binomial by squaring each term and then doubling the product of the two distinct terms

When we have an expression such as \((c + \frac{1}{c})^2\), we recognize that \(c\) and \(\frac{1}{c}\) play the roles of \(a\) and \(b\), respectively. By applying our identity, we can break down the binomial into three terms by:

• Squaring the first term: \(c^2\)
• Multiplying the terms together and doubling it: \(2 \, (c \cdot \frac{1}{c}) = 2\)
• Squaring the second term: \((\frac{1}{c})^2 = \frac{1}{c^2}\)

This transforms our binomial into the expanded form, which makes further steps like simplification straightforward.

Squaring a Binomial

Squaring a binomial is a specific operation where you raise an expression made up of two terms to the power of two. This operation is fundamental in algebra, allowing us to transform and manipulate polynomial expressions. When we square a binomial, we use the formula \((a+b)^2 = a^2 + 2ab + b^2\).

In our exercise, we square the whole binomial \(\left(c + \frac{1}{c}\right)^2\). This involves three main operations:

• Compute \(c^2\)
• Calculate the middle term \(2ab\), which simplifies to \(2\) because \(c \cdot \frac{1}{c} = 1\)
• Finish with \((\frac{1}{c})^2\), leading to \(\frac{1}{c^2}\)

The process of squaring is straightforward but requires attentiveness to correctly apply the identity and gather all parts of the expansion into a coherent whole.

Simplifying Expressions

Simplifying expressions involves combining and condensing terms to make the expression easier to understand or work with. The goal is to make expressions as simple as possible while keeping them equivalent to the original form. For algebraic expressions, this often means performing operations like addition, subtraction, multiplication, or division on coefficients or like terms.

In the context of the expanded binomial \(c^2 + 2 + \frac{1}{c^2}\), we simplify it down to its most compact form by ensuring all terms are gathered without any further reduction possible. Simplification reveals underlying patterns or values that might not be immediately visible in expanded form.

For our example:

• Combine constants or similar terms when applicable.
• Ensure fractions are properly managed to keep the expression neat.
• Always check results against the original expression to confirm equivalence.

Simplification can often be the final step in solving an equation, offering a neat and readable form which represents the original problem.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. They serve as the foundation of algebra, representing a wide range of mathematical scenarios. Understanding how to manipulate these expressions is key to solving algebraic equations and problems.

An algebraic expression might be as simple as a single term, like \(x\), or more complex, involving several variables and operations as seen with \(c^2 + 2 + \frac{1}{c^2}\). These expressions are versatile, allowing for various manipulations:

• Adding or subtracting expressions
• Multiplying or dividing terms
• Factoring or expanding components

The expression \(\left(c + \frac{1}{c}\right)^2\) illustrates this versatility. Starting as a simple binomial square, it expands and simplifies using algebraic identities into a less complex form: \(c^2 + 2 + \frac{1}{c^2}\). This process exemplifies how algebra facilitates transforming and understanding mathematical relationships.