Problem 12 - Algebra 2

Question

Expand each power. $$ (m+n)^{4} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Binomial Theorem

The binomial theorem provides a way to expand a binomial raised to a power. It 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.

Step 2: Apply the Binomial Theorem

Using the binomial theorem, plug in \(a = m\), \(b = n\), and \(n = 4\).\[(m+n)^4 = \sum_{k=0}^{4} \binom{4}{k} m^{4-k} n^{k}\]

Step 3: Calculate the Binomial Coefficients

Calculate each binomial coefficient for \(k = 0, 1, 2, 3, 4\):- \( \binom{4}{0} = 1\) - \( \binom{4}{1} = 4\) - \( \binom{4}{2} = 6\) - \( \binom{4}{3} = 4\) - \( \binom{4}{4} = 1\)

Step 4: Expand the Binomial Expression

Plug these coefficients into the binomial expansion formula:\[(m+n)^4 = 1 \cdot m^4 + 4 \cdot m^3 \cdot n + 6 \cdot m^2 \cdot n^2 + 4 \cdot m \cdot n^3 + 1 \cdot n^4\]

Step 5: Simplify the Expression

Combine and simplify the expression:\[(m+n)^4 = m^4 + 4m^3n + 6m^2n^2 + 4mn^3 + n^4\]

Key Concepts

Binomial Expansion

The binomial expansion is a fundamental concept in algebra used to express the powers of a binomial as a sum of terms. The binomial expansion follows a systematic pattern governed by the binomial theorem. When you raise a binomial, such as \((m+n)\), to a power like 4, it generates a series of terms combined in a specific way.For \((m+n)^4\), the binomial expansion provides an expanded form that consists of terms with varying powers of \(m\) and \(n\). These terms are formed by applying the binomial theorem:

• Each term in the expansion is a unique combination of \(m\) and \(n\).
• The exponents of \(m\) and \(n\) in each term add up to 4, which is the power of the binomial.
• The number of terms in the expansion is one more than the power of the binomial, making it 5 in this case.

Understanding the layout and structure of the expansion helps in breaking down the expression step-by-step.

Binomial Coefficient

The binomial coefficient is crucial in determining the weights of each term in a binomial expansion. Represented as \( \binom{n}{k} \), it is a mathematical way to select \(k\) items from \(n\) items without consideration of order. In our scenario:

• The binomial coefficients for \((m+n)^4\) are calculated as follows:\(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), \(\binom{4}{4}\)
• When computed, these coefficients are 1, 4, 6, 4, and 1 respectively.
• These coefficients determine how each product of \(m\) and \(n\) in the expansion is scaled.

These coefficients can be found directly from Pascal's Triangle or using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Using the coefficients correctly is vital in obtaining the accurate expanded expression.

Algebraic Expressions

Algebraic expressions are composed of variables, numbers, and operations combined in a meaningful way. In the context of the binomial expansion, the algebraic expression \((m+n)^4\) represents a combination of terms brought together through addition.In the expanded form of \((m+n)^4\),

• Each term is an algebraic expression itself, consisting of products of variables raised to different powers.
• The expression is simplified by calculating and combining like terms, which involves applying arithmetic operations to consolidate them.
• The final simplified form of the expansion results in a neatly organized series of terms: \(m^4 + 4m^3n + 6m^2n^2 + 4mn^3 + n^4\).

Handling algebraic expressions involves understanding how to manage powers, coefficients, and variables collaboratively to maintain the integrity of the mathematical expression.