Problem 36 - 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: Understand the Problem

We are given the expression \( \left(c + \frac{1}{c}\right)^2 \). Our task is to expand and simplify this expression.

Step 2: Apply the Formula for Squaring a Binomial

The square of a binomial \((a+b)^2\) is given by the formula: \( a^2 + 2ab + b^2 \). In our problem, let \( a = c \) and \( b = \frac{1}{c} \). Substitute these into the formula.

Step 3: Calculate \(a^2\)

Calculate \( c^2 \) since \( a = c \): \( c^2 = c^2 \).

Step 4: Calculate \(2ab\)

Substitute \( a = c \) and \( b = \frac{1}{c} \) into \( 2ab \): \( 2 \times c \times \frac{1}{c} = 2 \times 1 = 2 \).

Step 5: Calculate \(b^2\)

Calculate \( b^2 \) where \( b = \frac{1}{c} \): \( \left(\frac{1}{c}\right)^2 = \frac{1}{c^2} \).

Step 6: Combine All Terms

Put all calculated parts together: \( c^2 + 2 + \frac{1}{c^2} \).

Step 7: Simplify the Expression

The expression \( c^2 + 2 + \frac{1}{c^2} \) is already simplified. Therefore, this is the final result.

Key Concepts

Binomial Expansion

Expanding a binomial involves breaking it down using the binomial theorem. The theorem provides a formula to express powers of binomials in a polynomial form. This is especially useful for expressions like \((a + b)^n\), where n is a non-negative integer. The expanded form includes terms of the form \(\binom{n}{k} a^{n-k} b^k\), where \(k\) ranges from 0 to \(n\) and \(\binom{n}{k}\) is a binomial coefficient.

Our specific problem requires us to expand \(\left(c + \frac{1}{c}\right)^2\). However, since \(n = 2\), we can directly apply the simpler identity for the square of a binomial, instead of using the full binomial theorem. This involves determining each term separately according to the formula, which makes the process straightforward.

Understanding binomial expansion equips students with the ability to tackle a wide range of algebraic problems effectively, employing a systematic approach to expressions of this form.

Squaring a Binomial

Squaring a binomial means multiplying the binomial by itself, resulting in a specific algebraic operation defined by the formula \((a + b)^2 = a^2 + 2ab + b^2\). This formula simplifies the process by breaking down the operation into manageable parts.

For example, in the expression \((c + \frac{1}{c})^2\), we identify \(a = c\) and \(b = \frac{1}{c}\). Using the squaring formula, we compute:

• \(a^2 = c^2\)
• \(2ab = 2 \times c \times \frac{1}{c} = 2\)
• \(b^2 = \left(\frac{1}{c}\right)^2 = \frac{1}{c^2}\)

Combining these results, the squared binomial simplifies to \(c^2 + 2 + \frac{1}{c^2}\). This method is efficient, ensuring that each part of the original binomial contributes to the final expanded expression.

By mastering the squaring of binomials, students gain a fundamental skill necessary for advanced algebraic manipulations.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression more digestible or compact. This involves combining like terms and reducing fractions wherever possible without changing the expression's value. In many cases, an expression can become quite complex after operations such as expansions, making simplification a crucial step.

In our specific example of \(c^2 + 2 + \frac{1}{c^2}\), each term is distinct and cannot be combined further, indicating that our expression is already simplified. Simplifying follows basic algebraic rules, like maintaining order of operations and recognizing values that can cancel.

Here are some general tips for simplifying expressions:

• Always look to combine similar terms.
• Simplify fractions by finding common denominators when necessary.
• Re-order the expression if it aids in clarity or further reduction.

Having a solid grasp of simplifying expressions not only aids in solving math problems but also improves overall mathematical literacy, ensuring clarity and accuracy.