Problem 41 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ (m+n)^{6} $$

Step-by-Step Solution

Answer

The expansion is \( m^6 + 6m^5n + 15m^4n^2 + 20m^3n^3 + 15m^2n^4 + 6mn^5 + n^6 \).

Step 1: Understanding Pascal's Triangle

Pascal's triangle is a triangular array of numbers with each row corresponding to the coefficients of the binomial expansion. The coefficients for the expansion of \( (a+b)^6 \) are found in the 6th row of Pascal's triangle (starting from row 0). The numbers in this row are 1, 6, 15, 20, 15, 6, 1.

Step 2: Writing the Binomial Expansion Formula

The expansion of \( (m+n)^6 \) is given by the formula: \[ (m+n)^6 = inom{6}{0}m^6n^0 + inom{6}{1}m^5n^1 + inom{6}{2}m^4n^2 + inom{6}{3}m^3n^3 + \]\[ inom{6}{4}m^2n^4 + inom{6}{5}m^1n^5 + inom{6}{6}m^0n^6 \]The coefficients (binomial coefficients) are obtained from Pascal's Triangle.

Step 3: Applying the Coefficients

Using the coefficients from Pascal's triangle, the expansion becomes:\[ 1 imes m^6 + 6 imes m^5n + 15 imes m^4n^2 + 20 imes m^3n^3 + \]\[ 15 imes m^2n^4 + 6 imes mn^5 + 1 imes n^6 \]

Step 4: Simplifying Each Term

The final expanded form of \( (m+n)^6 \) is:\[ m^6 + 6m^5n + 15m^4n^2 + 20m^3n^3 + 15m^2n^4 + 6mn^5 + n^6 \]

Key Concepts

Binomial Expansion

The binomial expansion is a method used to express the powers of sums. This is especially useful when you have expressions in the form of \((a + b)^n\). The goal is to "expand" this expression to find all individual terms. Each term in the expansion is a product of a coefficient, a power of \(a\), and a power of \(b\). For our example, \((m + n)^6\), we want to expand this into a sum of terms that are easier to handle and understand.

• When you expand, you multiply the terms \(m\) and \(n\) by different combinations and use coefficients to scale these terms appropriately.
• The expansion relies on using binomial coefficients, which are numbers that scale each term in the expansion.
• The number of terms in the expansion is \(n + 1\), where \(n\) is the power of the binomial expression.

Mastering binomial expansion helps simplify algebraic expressions and solves problems involving powers of binomials efficiently.

Binomial Coefficients

Binomial coefficients play a crucial role in binomial expansion. These coefficients are values that depend on the number of terms (n) and the particular term number (k) in the expansion.
Formulaically, a binomial coefficient is represented by \(\binom{n}{k}\). This notation stands for the number of ways to choose \(k\) items from \(n\) items without regard to order.

In simpler terms, it tells us how many ways we can get a particular power combination of two variables. Considering our example of \((m+n)^6\):

• The coefficient 1 in \(m^6\) is obtained from \(\binom{6}{0}\).
• The coefficient 6 in \(m^5n\) comes from \(\binom{6}{1}\)
• Each coefficient matches a point from Pascal's Triangle, a clever tool to find binomial coefficients easily.

Understanding binomial coefficients allows one to determine the size of each component term in an expansion, simplifying the complexity of calculations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators such as addition and multiplication. When dealing with expressions like \((m+n)^6\), each variable and operation describes a component of a larger algebraic structure.

• These expressions can encode complex mathematical ideas, and binomial expansion is one way to "unpack" those ideas into simpler parts.
• By simplifying an expression like \((m+n)^6\), we find a straightforward polynomial with terms like \(m^6\) or \(15m^4n^2\).
• The goal is to take an expression that looks complicated and find a version that looks like a long addition problem.

Breaking down algebraic expressions into expanded forms enhances comprehension and allows for easier manipulation in solving equations or modeling real-world situations.