Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Binomial Theorem

The Binomial Theorem states that for any positive integer \( n \), \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\), where \( \binom{n}{k} \) is a binomial coefficient.

Step 2: Assign Values to Variables

In the expression \((3a + 2b)^3\), identify \(x\) and \(y\). Here, \(x = 3a\), \(y = 2b\) and \(n = 3\).

Step 3: Set Up the Expansion

Use the Binomial Theorem formula: \((3a + 2b)^3 = \sum_{k=0}^{3} \binom{3}{k} (3a)^{3-k} (2b)^k\). We will calculate each term of this sum.

Step 4: Calculate Each Term

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

Step 5: Combine the Terms

Add the terms calculated in Step 4 to form the expanded expression:\[(3a + 2b)^3 = 27a^3 + 54a^2b + 36ab^2 + 8b^3\].

Key Concepts

Binomial Expansion

The binomial expansion is a way of expressing expressions that are raised to a power, specifically binomials, which are expressions with two terms. By using the Binomial Theorem, you can efficiently expand expressions like i.e., \[ (3a + 2b)^3 \] into a sum of terms that are easier to manage. To use binomial expansion, you essentially replace the binomial with a series of terms that involve coefficients, powers of the first term, and powers of the second term.

• Each term in the expansion has a specific coefficient and involves powers of both components of the binomial.
• The sum of the exponents in each term always equals the original power to which the binomial was raised.

Understanding binomial expansion is crucial for simplifying complex algebraic equations and can significantly reduce the amount of manual calculation required.

Binomial Coefficients

Binomial coefficients are a key component of expanding binomials using the Binomial Theorem. They are the numbers \( \binom{n}{k} \) which can be found in Pascal's Triangle and represent the coefficients of the terms in the expansion.

• The binomial coefficient \( \binom{n}{k} \) is defined as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
• They help determine how many ways you can choose \(k\) elements from \(n\) without regard to order.

For example, in the term \( \binom{3}{1} (3a)^2 (2b)^1 \),the binomial coefficient \( \binom{3}{1} \) is equal to 3, which indicates that there are 3 such terms in the expansion. It's important to note that these coefficients allow the symmetrical combinations of powers from both terms to appear in sequence.

Algebra Expansion

Algebra expansion involves multiplying out expressions to express them in a simplified, longer form. When you apply this concept to problems like \( (3a + 2b)^3 \), it means taking each term one by one, applying the powers and coefficients, and writing them as a summation of various terms.

• This process requires careful handling of coefficients, product rules, and ensuring that all terms are accounted for.
• When you expand \( (3a + 2b)^3 \), an equivalent expression is \( 27a^3 + 54a^2b + 36ab^2 + 8b^3 \).
• Each term results from systematically applying the exponential powers to each part of the binomial and multiplying by the relevant coefficients.

Mastering algebra expansion is helpful in a range of mathematical tasks, from simplifying equations to solving mathematical problems.