Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Write the binomial expansion for each expression. $$\left(m+n^{2}\right)^{4}$$

Step-by-Step Solution

Answer

The expansion is \(m^{4} + 4m^{3}n^{2} + 6m^{2}n^{4} + 4mn^{6} + n^{8}\).

Step 1: Identify the Expression Form

To write the binomial expansion, start by recognizing the expression as \((a + b)^{n}\). In the given expression \((m + n^{2})^{4}\), \(a = m\), \(b = n^{2}\), and the exponent \(n = 4\).

Step 2: Understand the Binomial Theorem

The binomial theorem states that \[(a + b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k,\] where \(\binom{n}{k}\) is the binomial coefficient defined as \[\binom{n}{k} = \frac{n!}{k!(n-k)!}.\]

Step 3: Calculate the Binomial Coefficients

Calculate the binomial coefficients for \(n = 4\): - \(\binom{4}{0} = 1\) - \(\binom{4}{1} = 4\) - \(\binom{4}{2} = 6\) - \(\binom{4}{3} = 4\) - \(\binom{4}{4} = 1\).

Step 4: Expand Using the Binomial Theorem

Substitute the values \(a = m\), \(b = n^{2}\), and the binomial coefficients into the binomial expansion formula:\[(m + n^{2})^{4} = \binom{4}{0} m^{4} (n^{2})^{0} + \binom{4}{1} m^{3} (n^{2})^{1} + \binom{4}{2} m^{2} (n^{2})^{2} + \binom{4}{3} m^{1} (n^{2})^{3} + \binom{4}{4} m^{0} (n^{2})^{4}.\]

Step 5: Simplify Each Term

Evaluate and simplify each term in the expansion:- \(\binom{4}{0} m^{4} (n^{2})^{0} = m^{4}\) - \(\binom{4}{1} m^{3} (n^{2})^{1} = 4m^{3}n^{2}\) - \(\binom{4}{2} m^{2} (n^{2})^{2} = 6m^{2}n^{4}\) - \(\binom{4}{3} m^{1} (n^{2})^{3} = 4mn^{6}\) - \(\binom{4}{4} m^{0} (n^{2})^{4} = n^{8}\).

Step 6: Write the Complete Binomial Expansion

Combine all the simplified terms to write the full expanded form of the expression:\[(m+n^{2})^{4} = m^{4} + 4m^{3}n^{2} + 6m^{2}n^{4} + 4mn^{6} + n^{8}.\]

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept in algebra that allows us to expand expressions of the form \((a + b)^{n}\). This powerful tool provides a way to break down complex binomial expressions into simpler terms. The theorem states that: \[(a + b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k},\]where \(\binom{n}{k}\) are the binomial coefficients. This formula can be used to expand any binomial raised to a power \(n\).

• "\(a\)" and "\(b\)" represent the terms of the binomial.
• "\(n\)" is the exponent applied to the binomial.

Understanding the binomial theorem helps simplify and solve numerous algebraic problems, as it breaks down the problem into a series of manageable terms.

Binomial Coefficients

At the heart of the binomial theorem are the binomial coefficients, denoted as \(\binom{n}{k}\). These coefficients play a crucial role in determining the weight of each term in a binomial expansion. Binomial coefficients are defined as: \[\binom{n}{k} = \frac{n!}{k!(n-k)!},\]where "\(!\)" symbolizes the factorial of a number, meaning you multiply all positive integers up to that number. For example, \(4!\) equals 24 because \(4 \times 3 \times 2 \times 1 = 24\). Apply these coefficients to the terms in a binomial expansion to ensure each term is properly scaled:

• The coefficient for the first term \(k=0\) is always 1, which means no multiplier other than the term itself.
• As \(k\) increases, compute \(\binom{n}{k}\) to determine the proportionate impact of each term.

These coefficients are symmetric, meaning \(\binom{n}{k} = \binom{n}{n-k}\), simplifying some calculations.

Exponent

Exponents are not just labels of repeated multiplication; they are pivotal to calculating powers of numbers and binomial expansions. In the context of binomial expansions, the exponent \(n\) influences how a binomial such as \((a + b)^{n}\) should be expanded. Here's why the exponent is important:

• It tells us how many times \(a + b\) is multiplied by itself.
• Each term in the expansion involves powers of the terms \(a\) and \(b\) up to \(n\).
• The exponent directly affects the number of terms in the expansion, determined to be \(n+1\) terms.

Understanding the impact of the exponent helps in visualizing and calculating each stage of the expansion correctly, leading to the accurate formation of terms.

Algebra

Algebra encompasses the study of symbols and the rules for manipulating these symbols. It provides the foundation for working with binomials, using functions like addition, multiplication, and especially, exponentiation. Binomial expansion is a prime application of algebraic principles. Algebra allows us to:

• Generalize arithmetic using variables.
• Break down complex expressions to simpler parts.
• Apply rules like distributive and associative laws to facilitate expansion.

Algebraic operations simplify calculations in binomial expansions as they help systematically apply multiplication and addition over multiple terms. Mastery of algebra helps in understanding and performing operations involved in the binomial theorem.