Problem 32 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ (m+n)^{3} $$

Step-by-Step Solution

Answer

(m+n)^3 = m^3 + 3m^2n + 3mn^2 + n^3.

Step 1: Identify the Row in Pascal's Triangle

Since you need to expand (m+n)^3 , look for the 4th row in Pascal's triangle (remember the topmost row is the 0th row). This row corresponds to [1, 3, 3, 1] , which will give the coefficients for the expansion.

Step 2: Write the General Binomial Expansion Formula

Recall that the expanded form of (a+b)^n is ∑_{k=0}^{n} C(n,k) * a^{n-k} * b^k , where C(n,k) are the coefficients of Pascal's triangle.

Step 3: Apply the Coefficients from Pascal's Triangle

Using the coefficients [1, 3, 3, 1] from step 1, apply these to the general formula: (m+n)^3 = 1*m^3 + 3*m^2*n + 3*m*n^2 + 1*n^3.

Step 4: Simplify Each Term

Calculate and simplify each term: 1*m^3 = m^3, 3*m^2*n = 3m^2n, 3*m*n^2 = 3mn^2, and 1*n^3 = n^3.

Step 5: Combine the Terms

Put all the terms together to get the final expanded expression: (m+n)^3 = m^3 + 3m^2n + 3mn^2 + n^3.

Key Concepts

Pascal's Triangle

Pascal's Triangle is a fascinating and highly useful mathematical tool. It takes the form of a triangular array, where each number is the sum of the two numbers directly above it. This triangle begins with a '1' at the top, known as the 0th row. The rows are numbered starting from zero, and each number in the row symbolizes the coefficients used in binomial expansion.

• The 0th row is just \( [1] \)
• The 1st row is \( [1, 1] \)
• The 2nd row is \( [1, 2, 1] \)
• The 3rd row is \( [1, 3, 3, 1] \)

When expanding a binomial expression with an exponent of n, you simply use the coefficients from the nth row of Pascal's Triangle. For expanding \((m+n)^3\), the 3rd row \( [1, 3, 3, 1] \) provides the coefficients that will be used.

Binomial Theorem

The Binomial Theorem is a powerful way of expanding expressions that are raised to a power. It states that for any integers a, b, and non-negative integer n, the expression \((a + b)^n\) can be expanded into a sum involving terms of the form \(C(n,k) \cdot a^{n-k} \cdot b^k\). Here, \(C(n,k)\) represents the binomial coefficient, which can be found in Pascal's Triangle.
Every term in this expansion consists of a binomial coefficient, a power of a, and a power of b. For example, in \((m+n)^3\), you'll have:

• The first term: \(1 \times m^3 \times n^0\)
• The second term: \(3 \times m^2 \times n^1\)
• The third term: \(3 \times m^1 \times n^2\)
• The fourth term: \(1 \times m^0 \times n^3\)

This expansion allows calculations without directly multiplying out the entire expression, making it particularly useful for dealing with higher powers.

Coefficient Calculation

Coefficient Calculation is an integral part of using the Binomial Theorem. Coefficients determine how each term in your binomial expansion will look. These coefficients can be calculated using combinations, but a more straightforward method is through Pascal's Triangle.

• Each row in Pascal's Triangle corresponds to the coefficients needed for binomial expansion with the same exponent.
• For an expansion like \((m+n)^3\), look at the 3rd row: \([1, 3, 3, 1]\).
• These numbers tell you how many of each type of term you'll have in your expansion.

To apply these coefficients effectively, they are multiplied by the terms as shown in the Binomial Theorem. For example, in \((m+n)^3\), the coefficient 3 tells you that each term of \(m^2 \cdot n\) and \(m \cdot n^2\) appears three times in the combined expanded expression. Understanding these coefficients simplifies the process and helps ensure each step of your expansion is accurate.