Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (3 a+2 b)^{3} $$

Step-by-Step Solution

Answer

The expansion of \((3a + 2b)^3\) is \(27a^3 + 54a^2b + 36ab^2 + 8b^3\).

Step 1: Identify Components

The binomial to be expanded is \((3a + 2b)^3\). The Binomial Theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). In this case, \(x = 3a\), \(y = 2b\), and \(n = 3\).

Step 2: Apply Binomial Theorem

Using the Binomial Theorem, we will expand the expression: \((3a + 2b)^3\). According to the theorem, it is equal to:\[\sum_{k=0}^{3} \binom{3}{k} (3a)^{3-k} (2b)^k\]

Step 3: Compute Individual Terms

We calculate each term of the sum:- For \(k=0\): \(\binom{3}{0} (3a)^3 (2b)^0 = 1 \cdot 27a^3 \cdot 1 = 27a^3\)- For \(k=1\): \(\binom{3}{1} (3a)^2 (2b)^1 = 3 \cdot 9a^2 \cdot 2b = 54a^2b\)- For \(k=2\): \(\binom{3}{2} (3a)^1 (2b)^2 = 3 \cdot 3a \cdot 4b^2 = 36ab^2\)- For \(k=3\): \(\binom{3}{3} (3a)^0 (2b)^3 = 1 \cdot 1 \cdot 8b^3 = 8b^3\)

Step 4: Sum All Terms

Add all the individual terms calculated in Step 3:\(27a^3 + 54a^2b + 36ab^2 + 8b^3\)

Key Concepts

Binomial Expansion

When faced with expanding a binomial expression like \((3a + 2b)^3\), the Binomial Theorem offers a systematic way to achieve this. The binomial expression consists of terms in the format \( (x+y)^n \), where \(x\) and \(y\) represent the individual terms of the binomial, and \(n\) is the power to which the binomial is raised. Using the theorem, each term in the expansion follows a pattern, guided by the formula:

• \((x + y)^n = \ sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

This formula incorporates several elements:

• \(\binom{n}{k}\): This is a binomial coefficient calculated using combinatorics, which we'll delve into shortly.
• \(x^{n-k}\) and \(y^k\): These represent the decreasing powers of \(x\) and the increasing powers of \(y\), effectively distributing the degree across each term in the expansion.

The methodical application of the binomial theorem ensures a consistent result each time, as we compute various terms until reaching \(n\). Understanding this theorem opens a gateway to tackling more complex algebraic expressions.

Combinatorics

A central component of the Binomial Theorem is the combinatorial aspect, specifically the binomial coefficient. It is represented mathematically as \(\binom{n}{k}\), commonly read as "n choose k". This coefficient represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order. The calculation of this coefficient is done using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Where "!" denotes a factorial, meaning the product of all positive integers up to that number. For instance, in expanding \((3a + 2b)^3\), when calculating each term, we refer to combinatorics for coefficients:

• \(\binom{3}{0} = 1\)
• \(\binom{3}{1} = 3\)
• \(\binom{3}{2} = 3\)
• \(\binom{3}{3} = 1\)

Combinatorics allows us to manage polynomials systematically by understanding how terms distribute across binomials. It is foundational for constructing the structure of the polynomial's expanded form.

Polynomial Algebra

Polynomial algebra plays a crucial role in expanding binomials and involves manipulating expressions based on the properties of polynomials. Once the binomial expression \((3a + 2b)^3\) has been expanded using the Binomial Theorem, each term becomes part of a larger polynomial. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

• The terms from our example: \(27a^3, 54a^2b, 36ab^2, \text{and } 8b^3\)

Concise understanding and manipulation of these polynomials are made attainable by recognizing:

• Degrees: The degree of a polynomial corresponds to the highest power of a variable present; here, our resulting polynomial is of degree 3.
• Terms: Each piece like \(27a^3\) is a term, contributing to the polynomial’s overall form.
• Coefficients: Numeric factors of terms, e.g., 27 in \(27a^3\), illustrate the weight each term contributes to the polynomial.

Engaging with polynomial algebra enables solving more extensive algebraic operations and equations. Mastery over it makes solving complex polynomial expressions less daunting.