Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((2+i)^{5}\)

Step-by-Step Solution

Answer

The simplified form of \((2+i)^5\) is \(-16 + i\).

Step 1: Use Binomial Theorem

Use the binomial theorem to expand the expression \((2+i)^5\). This theorem states that \((a+b)^n = \sum_{k=0}^{n} {n \choose k}a^{n-k}b^k\). So, applying the theorem to \((2+i)^5\), we get: \(2^5+ {5 \choose 1}2^4 \cdot i+ {5 \choose 2}2^3 \cdot i^2+ {5 \choose 3}2^2 \cdot i^3+ {5 \choose 4}2 \cdot i^4+ i^5\).

Step 2: Evaluate the Expressions

Evaluate the expression obtained from Step 1: \(32+ {5 \choose 1}\cdot 32 \cdot i+ {5 \choose 2}\cdot 8 \cdot i^2+ {5 \choose 3}\cdot 4 \cdot i^3+ {5 \choose 4}2\cdot i^4+ i^5\). This becomes: \(32+ 160i - 80 - 160i + 32 + i^5\).

Step 3: Simplify the Complex Number

Simplify the expression obtained from Step 2: Since \(i^2 = -1, i^3 = -i, i^4 = 1\) and \(i^5 = i\), the expression becomes: \(-16 + i\) after simplification.

Key Concepts

Complex Number Expansion

Understanding the expansion of complex numbers through the binomial theorem provides a foundation for working with powers of complex numbers. The process involves applying the theorem to separate the real and imaginary parts of the complex number.

When expanding \( (a+bi)^n \) for a complex number \( a+bi \) where \( i \) represents the imaginary unit \( \sqrt{-1} \) and \( n \) is a positive integer, the expansion will result in a combination of terms with real and imaginary coefficients. The alternating powers of \( i \) (like \( i^2 = -1, i^3 = -i \) etc.) are critical since they affect the sign and reality of each term.

The challenges often involve managing sign changes and applying the correct power of \( i \) during the expansion. Simplification then, is not only about calculating coefficients but also about understanding how the properties of \( i \) influence the final expression.

Binomial Expansion

The binomial expansion is a mathematical method used to express the power of a sum in terms of its individual terms. It follows the binomial theorem, which succinctly presents the expansion as a sum of terms in the form \( {n \choose k}a^{n-k}b^k \) where \( a \) and \( b \) are any numbers, and \( k \) ranges from 0 to \( n \).

To apply the binomial theorem effectively to complex numbers, one must be comfortable with calculating binomial coefficients, which can also be found using Pascal's triangle, and raising numbers to powers. These coefficients dictate the multiplicative factors in front of each term in the expansion.

Simplifying Complex Expressions

Simplifying complex expressions is a critical skill that involves reducing the expression to its most basic form. After the binomial expansion, it's essential to combine like terms—those that have the same powers of \( i \)—and apply the properties of \( i \) to further simplify the expression.

For instance, using the identity \( i^2 = -1 \) lends a hand in converting even powers of \( i \) to real numbers. Odd powers typically lead to imaginary terms, with \( i^3 \) simplifying to \( -i \) and so on, following the pattern. This simplification is vital to present the final answer in the standard form \( a + bi \) where \( a \) and \( b \) are real numbers.

Pascal's Triangle

Pascal's triangle is a geometric representation of binomial coefficients arranged in a triangle. Each row of the triangle corresponds to the coefficients of the binomial expansion of \( (a+b)^n \), starting with \( n = 0 \) at the top of the triangle.

The triangle is constructed by starting with a ‘1’ at the top, and each subsequent number is the sum of the two numbers directly above it. For example, the fifth row of Pascal’s triangle (ignoring the initial 1) gives the coefficients for the binomial expansion of \( (a+b)^4 \) as 1, 4, 6, 4, 1. A clear understanding of Pascal's triangle simplifies the process of finding the coefficients required for expanding binomials and is particularly handy when dealing with higher powers, like in the given exercise \( (2+i)^5 \) where the coefficients can be quickly identified from the fifth row of the triangle.