Expanding (c - frac{1}{c}) using the Binomial Theorem

The binomial theorem is an essential tool in expanding expressions of the form ((a b)^n). In this article, we will explore how to expand (c - frac{1}{c}) to the fourth power using the binomial theorem. This process will be outlined with detailed steps and examples to ensure a comprehensive understanding.

Introduction to the Binomial Theorem

The binomial theorem states that for any non-negative integer (n) and any real numbers (a) and (b), the expansion of (a b) raised to the power (n) is given by:

[ (a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k ]

Applying the Binomial Theorem to Expand (c - frac{1}{c})

To expand (c - frac{1}{c}) to the fourth power using the binomial theorem, we can express the problem in a form that fits the binomial expansion formula.

Let's start by substituting (c) for (a) and (-frac{1}{c}) for (b). Then the expression becomes:

[ left(c - frac{1}{c}right)^4 ]

Step-by-Step Expansion

Using the binomial theorem, the expansion can be written as:

[ left(c - frac{1}{c}right)^4 sum_{k0}^{4} binom{4}{k} c^{4-k} left( -frac{1}{c} right)^k ]

Simplifying the terms inside the summation:

[ left(c - frac{1}{c}right)^4 binom{4}{0} c^4 left(-frac{1}{c}right)^0 binom{4}{1} c^3 left(-frac{1}{c}right)^1 binom{4}{2} c^2 left(-frac{1}{c}right)^2 binom{4}{3} c^1 left(-frac{1}{c}right)^3 binom{4}{4} c^0 left(-frac{1}{c}right)^4 ]

Focusing on each term in the summation:

[ binom{4}{0} c^4 left(-frac{1}{c}right)^0 c^4 cdot 1 c^4 ] [ binom{4}{1} c^3 left(-frac{1}{c}right)^1 4 c^3 cdot -frac{1}{c} -4 c^2 ] [ binom{4}{2} c^2 left(-frac{1}{c}right)^2 6 c^2 cdot frac{1}{c^2} 6 ] [ binom{4}{3} c^1 left(-frac{1}{c}right)^3 4 c cdot -frac{1}{c^3} -frac{4}{c^2} ] [ binom{4}{4} c^0 left(-frac{1}{c}right)^4 1 cdot frac{1}{c^4} frac{1}{c^4} ]

Combining all the terms, we get:

[ left(c - frac{1}{c}right)^4 c^4 - 4 c^2 6 - frac{4}{c^2} frac{1}{c^4} ]

Alternative Formulation

Another way to express the same expansion is to rewrite the expression as:

[ left( frac{c^2 - 1}{c} right)^4 ]

Expanding this form, we get:

[ left( frac{c^2 - 1}{c} right)^4 frac{(c^2 - 1)^4}{c^4} ]

Expanding the numerator using the binomial theorem:

[ (c^2 - 1)^4 sum_{k0}^{4} binom{4}{k} (c^2)^{4-k} (-1)^k ]

Simplifying again:

[ (c^2 - 1)^4 c^8 - 4c^6 6c^4 - 4c^2 1 ]

Thus, the full expansion is:

[ left( frac{c^2 - 1}{c} right)^4 frac{c^8 - 4c^6 6c^4 - 4c^2 1}{c^4} c^4 - 4c^2 6 - frac{4}{c^2} frac{1}{c^4} ]

Conclusion

In conclusion, the process of expanding (c - frac{1}{c}) to the fourth power using the binomial theorem involves several steps, including rewriting the expression for clarity. The final result is a polynomial expression that accurately represents the expansion. Understanding this process is crucial for solving various algebraic problems and working with more complex mathematical expressions.