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The Binomial Theorem is a fundamental principle in algebra that allows us to expand expressions that are raised to a power. It is particularly useful when dealing with polynomial expressions. When you come across an expression like \( (a + b)^n \), where \( n \) is a non-negative integer, the binomial theorem can be applied to expand it into a sum involving terms of the form \( a^{n-k}b^k \).

The general formula for binomial expansion is given by:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]Here, \( \binom{n}{k} \) represents the binomial coefficient, which can be calculated as \( \frac{n!}{k!(n-k)!} \), with \( ! \) denoting a factorial. This theorem is particularly handy when dealing with complex numbers, as it enables us to systematically expand and simplify powers of complex binomials.

The standard form of a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). In this form, \( a \) represents the real part, while \( bi \) represents the imaginary part of the complex number.

When writing the result of operations involving complex numbers, like squaring a complex number as shown in the example \( (5-4i)^2 \), it is essential to present the final answer in the standard form. This often involves simplification where the powers of \( i \) are resolved, and like terms are combined. The significance of the standard form lies in its simplicity and the ease with which it can be used in further arithmetic operations or visual representations on the complex plane.

Simplifying complex expressions involves a series of algebraic manipulations that lead to a standard form representation of the complex number. While working through these simplifications, it is important to consider properties of imaginary units and to ensure that all powers of \( i \) are appropriately dealt with.

To simplify complex expressions:

• Expand using algebraic identities, such as the binomial theorem.
• Apply the property that \( i^2 = -1 \) to resolve powers of \( i \).
• Combine like terms to consolidate the real and imaginary parts of the number.

Through these methods, complex expressions can be systematically broken down and rewritten in their simplest form. For instance, squaring \( (5 - 4i) \) requires expanding the expression and dealing with \( i^2 \) appropriately, finally resulting in \( 9 - 40i \) which can be easily interpreted and utilized for further calculations.