Problem 19 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ (m+2)^{3} $$

Step-by-Step Solution

Answer

The expanded form is \\( m^3 + 6m^2 + 12m + 8 \\\).

Step 1: Identify Components of the Binomial

The binomial theorem allows us to expand expressions of the form \( (a+b)^n \). In our expression, \( (m+2)^3 \), we identify \( a = m \), \( b = 2 \), and \( n = 3 \).

Step 2: Apply the Binomial Theorem Formula

The binomial theorem states that \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). We'll apply this formula to \( (m+2)^{3} \) with \( n = 3 \).

Step 3: Calculate Binomial Coefficients

For each term \( k \) from 0 to 3, calculate the binomial coefficient \( \binom{3}{k} \) using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). The coefficients are \( \binom{3}{0} = 1 \), \( \binom{3}{1} = 3 \), \( \binom{3}{2} = 3 \), \( \binom{3}{3} = 1 \).

Step 4: Expand the Expression using Each Term

Now substitute each term into the expression: \( (m+2)^3 = \binom{3}{0}m^{3}2^{0} + \binom{3}{1}m^{2}2^{1} + \binom{3}{2}m^{1}2^{2} + \binom{3}{3}m^{0}2^{3} \). Simplifying each term, we get: \( 1 \cdot m^3 \cdot 1 + 3 \cdot m^2 \cdot 2 + 3 \cdot m \cdot 4 + 1 \cdot 8 \).

Step 5: Simplify the Final Expression

Calculate and combine the terms: \( m^3 + 6m^2 + 12m + 8 \). This is the expanded form of \( (m+2)^3 \).

Key Concepts

Binomial Coefficients

In algebra, binomial coefficients play a significant role in expanding polynomial expressions raised to a power. They are represented as \( \binom{n}{k} \) and are essential in the binomial theorem, helping determine the coefficients for each term in the expansion.

• The general formula is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) represents the factorial of \( n \), \( k! \) the factorial of \( k \), and \( (n-k)! \) the factorial of \( (n-k) \).
• These coefficients show how many ways we can choose \( k \) elements from a set of \( n \) elements and are commonly used to calculate combinations.

For example, in the expansion of the expression \( (m+2)^3 \), the coefficients are 1, 3, 3, and 1 considering all the combinations for \( k = 0, 1, 2, \) and \( 3 \). Understanding these coefficients helps recognize patterns and calculate terms effectively.

Polynomial Expansion

Polynomial expansion involves breaking down expressions like \( (a+b)^n \) into simpler, additive forms. By applying the binomial theorem, any binomial expression can be expressed as a sum of terms.

• Each term in the expansion of \( (a+b)^n \) is structured as \( \binom{n}{k} a^{n-k} b^k \), where \( k \) iterates from 0 to \( n \).
• This structure is rooted in understanding both algebraic patterns and the power of binomials, allowing predictions about the form of various polynomial expressions.

In our example, \( (m+2)^3 \) expands systematically using these patterns: the first term being \( m^3 \), and the last \( 8 \), with each successive term's power of \( m \) decreasing and the power of 2 increasing.

Factorials

The concept of factorials, denoted by the symbol \(!\), forms a foundation in calculating binomial coefficients. Factorials are products of a sequence of descending natural numbers.

• For any positive integer \( n \), the factorial \( n! \) is calculated as \( n \times (n-1) \times (n-2) \times \, ... \, \times 1 \).
• Special factorial values include 0! which is defined as 1.

While computing binomial coefficients such as \( \binom{3}{2} = \frac{3!}{2!1!} \), the factorials simplify to provide the precise number of combinations needed, in this case, 3.

Algebra

Algebra is the overarching mathematical language used to manipulate symbols and numbers. It provides tools and rules to solve equations and expand expressions like binomials.

• Understanding algebra involves recognizing patterns, learning how to rearrange equations, and knowing how to identify components in expressions such as \( (m+2)^3 \).
• It requires the use of operations like addition, multiplication, and exponentiation.

Using algebraic principles, we not only expand expressions but also simplify them by combining like terms and organizing them in a simpler form, as seen in our final expression \( m^3 + 6m^2 + 12m + 8 \). These practices enhance problem-solving skills and logical thinking.