Problem 26 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ (3 a-2 b)^{5} $$

Step-by-Step Solution

Answer

The expanded form is \(243a^5 - 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 - 32b^5\).

Step 1: Understand the Binomial Theorem

The binomial theorem provides a formula to expand expressions of the form \((x + y)^n\). It states: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is a binomial coefficient.

Step 2: Identify Components

Identify the components for the binomial theorem: for \((3a - 2b)^5\), let \(x = 3a\), \(y = -2b\), and \(n = 5\). These values will be used in the theorem.

Step 3: Calculate Each Term Using the Binomial Theorem

Using the binomial theorem, calculate each term by substituting into the formula: \((3a - 2b)^5 = \sum_{k=0}^{5} \binom{5}{k} (3a)^{5-k} (-2b)^k\). Calculate each term for \(k = 0, 1, 2, 3, 4, 5\).

Step 4: Compute Binomial Coefficients

For each term, compute the binomial coefficient \(\binom{5}{k}\). The coefficients are 1, 5, 10, 10, 5, and 1 for \(k = 0, 1, 2, 3, 4, 5\), respectively.

Step 5: Calculate Individual Terms

Compute the individual terms: - For \(k = 0\): \(\binom{5}{0} (3a)^5 (-2b)^0 = 1 \,\cdot\, 243a^5 \,\cdot\, 1 = 243a^5\)- For \(k = 1\): \(\binom{5}{1} (3a)^4 (-2b)^1 = 5 \,\cdot\, 81a^4 \,\cdot\, (-2b) = -810a^4b\)- For \(k = 2\): \(\binom{5}{2} (3a)^3 (-2b)^2 = 10 \,\cdot\, 27a^3 \,\cdot\, 4b^2 = 1080a^3b^2\)- For \(k = 3\): \(\binom{5}{3} (3a)^2 (-2b)^3 = 10 \,\cdot\, 9a^2 \,\cdot\, (-8b^3) = -720a^2b^3\)- For \(k = 4\): \(\binom{5}{4} (3a)^1 (-2b)^4 = 5 \,\cdot\, 3a \,\cdot\, 16b^4 = 240ab^4\)- For \(k = 5\): \(\binom{5}{5} (3a)^0 (-2b)^5 = 1 \,\cdot\, 1 \,\cdot\, (-32b^5) = -32b^5\).

Step 6: Combine All Terms

Combine all the calculated terms to express the entire expansion:\((3a - 2b)^5 = 243a^5 - 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 - 32b^5\).

Key Concepts

Binomial Expansion

Binomial expansion is a mathematical process used to express the power of a binomial expression, such as \((x + y)^n\), in a simplified form. This process involves converting the power into a sum of terms using the binomial theorem. The binomial theorem provides a systematic method for expansion by incorporating binomial coefficients along with the components raised to specific powers.

Understanding binomial expansion makes it possible to handle complex algebraic expressions more easily, especially when dealing with large powers. This is because direct computation of large powers is often impractical.

• The binomial theorem formula is given as: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\).
• Each term in the expansion consists of a binomial coefficient, a power of the first component \(x\), and a power of the second component \(y\).

In practical applications, you break down each component based on their individual powers in each term. This is crucial for simplifying or solving more complex algebraic problems.

Binomial Coefficients

Binomial coefficients are key elements in binomial expansion. They are denoted as \(\binom{n}{k}\) and are read as 'n choose k'. These coefficients determine how many ways you can choose \(k\) elements from a set of \(n\) elements, which is mathematically calculated as \(\frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial, meaning the product of all positive integers up to that number.

Binomial coefficients play a pivotal role in determining the multipliers for each term in the binomial expansion.

• For example, in \((3a - 2b)^5\), the coefficients \(\binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \ldots, \binom{5}{5}\) are calculated to get 1, 5, 10, 10, 5, and 1.
• These coefficients are then used in conjunction with the powers of the individual components to find each term.

Understanding and calculating these coefficients accurately ensures the correct expansion and simplification of a binomial expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the context of the binomial theorem, these expressions are specifically structured in the form of a binomial, which includes two distinct terms.

For the expression \((3a - 2b)^5\), the terms \(3a\) and \(-2b\) are raised to the 5th power as per the binomial theorem.

• The expression is treated as a sum or difference of different terms, which interact according to the theorem.
• In each expansion term, variables such as \(a\) and \(b\) are raised to specific powers, which, when combined, form a new algebraic expression.

Thus, by breaking down complex algebraic expressions using methods like binomial expansion, students can gain deeper insights into how variables and numbers interact in higher powers. This ability is not only crucial in algebra but also in many other areas of mathematics and applied sciences.