Problem 43 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression \((3a - 2b)^5\) can be expanded using the binomial theorem as: \[ 243a^5 + 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 + 32b^5 \]

Step 1: Identify n, a, and b

In our case, \(n = 5\), \(a = 3a\), and \(b = -2b\).

Step 2: Write the binomial expansion formula

\((3a - 2b)^5 = \sum_{k=0}^{5} {5 \choose k} (3a)^{5-k}(-2b)^k\)

Step 3: Expand using the binomial coefficients

For each value of k from 0 to 5, we calculate the binomial coefficient \({5 \choose k}\), and plug it into the formula: \((3a - 2b)^5 = {5 \choose 0}(3a)^5(-2b)^0 + {5 \choose 1}(3a)^4(-2b)^1 + {5 \choose 2}(3a)^3(-2b)^2 + {5 \choose 3}(3a)^2(-2b)^3 + {5 \choose 4}(3a)^1(-2b)^4 + {5 \choose 5}(3a)^0(-2b)^5\)

Step 4: Simplify each term

Now, simplify the powers of \(3a\) and \(-2b\) and calculate the binomial coefficients: \((3a - 2b)^5 = 1(243a^5)(1) - 5(81a^4)(-2b) + 10(27a^3)(4b^2) - 10(9a^2)(-8b^3) + 5(3a)(16b^4) - 1(1)(-32b^5)\)

Step 5: Calculate the final coefficients

Combine the coefficients in each term: \((3a - 2b)^5 = 243a^5 + 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 + 32b^5\) So the final expanded expression of \((3a - 2b)^5\) is: \[ 243a^5 + 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 + 32b^5 \]

Key Concepts

Polynomial Expansion

Polynomial expansion is a method to express a binomial raised to a power as a sum of terms. In this case, we are expanding \((3a - 2b)^{5}\). Using the Binomial Theorem, we break this into multiple terms that are easier to understand. The expanded form not only provides each term individually but also makes calculations simpler and analysis more straightforward. This idea helps us see how combinations of terms multiply and adds to the complexity of expressions. It's a foundational tool for tackling more complex algebraic problems involving polynomials.

• Converts a binomial into numerous terms.
• Clarifies the structure and relationships between terms.
• Makes complex expressions manageable.

Binomial Coefficients

Binomial coefficients are special numbers that occur in the expansion of a binomial raised to a power. Represented as \({n \choose k}\), these coefficients tell us how many unique ways we can select \(k\) elements from \(n\) elements without considering the order.

• Formula: \({n \choose k} = \frac{n!}{k!(n-k)!}\).
• Appear in the binomial expansion of \((a + b)^n\).
• Help determine the weight of each term in expansion.
• Also found in Pascal's Triangle.

Binomial coefficients give shape to the distribution of terms in polynomial expansion. They ensure that we account for all possibilities when different terms are combined.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating them. It's about finding unknown values and understanding relationships through equations. In our exercise, algebra helps us manipulate the terms in the binomial expansion \((3a - 2b)^5\).

• Involves variables like \(a\) and \(b\) representing numbers.
• Uses operations like addition, subtraction, multiplication, and exponentiation.
• Helps solve equations and understand expressions dynamically.

Algebra's power in simplifying expressions means it's invaluable in expanding polynomials efficiently and correctly.

Powers and Exponents

Powers and exponents are fundamental in algebra. They describe how many times to multiply a number by itself. In our binomial expansion of \((3a - 2b)^5\), powers and exponents are crucial.

• Express repeated multiplication succinctly: \(a^n\) means \(a \times a \times ... \times a\) (\(n\) times).
• Exponent rules help in simplifying and combining terms.
• Vital for distributing powers across both variables, \(a\) and \(b\).

Understanding how to manipulate powers and exponents allows us to reduce complex multiplication into manageable steps, making it easier to handle and interpret algebraic expressions.