Problem 32 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$(4 a-5 b)^{5}$$

Step-by-Step Solution

Answer

(4a-5b)^5 = 1024a^5 - 2560a^4b + 4000a^3b^2 - 3000a^2b^3 + 1250ab^4 - 3125b^5

Step 1: Recognize the Binomial Theorem

The binomial theorem states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Here, the expression is \((4a - 5b)^5\), which suggests \(x = 4a\), \(y = -5b\), and \(n = 5\).

Step 2: Apply the Binomial Coefficient

For each term in the expansion, the coefficient is given by \(\binom{5}{k}\), where \(k\) goes from 0 to 5. Calculate the binomial coefficients for each value of \(k\).

Step 3: Determine Each Term's Form

Substitute \(x = 4a\) and \(y = -5b\) into the formula. Each term in the expansion will take the form: \(\binom{5}{k} (4a)^{5-k} (-5b)^k\).

Step 4: Calculate Each Term

Now, compute each term by replacing \(k\) from 0 to 5:- For \(k = 0\): \(\binom{5}{0}(4a)^5(-5b)^0 = 1024a^5\)- For \(k = 1\): \(\binom{5}{1}(4a)^4(-5b)^1 = -2560a^4b\)- For \(k = 2\): \(\binom{5}{2}(4a)^3(-5b)^2 = 4000a^3b^2\)- For \(k = 3\): \(\binom{5}{3}(4a)^2(-5b)^3 = -3000a^2b^3\)- For \(k = 4\): \(\binom{5}{4}(4a)^1(-5b)^4 = 1250ab^4\)- For \(k = 5\): \(\binom{5}{5}(4a)^0(-5b)^5 = -3125b^5\)

Step 5: Write the Full Expansion

Combine all the terms calculated: \[(4a-5b)^5 = 1024a^5 - 2560a^4b + 4000a^3b^2 - 3000a^2b^3 + 1250ab^4 - 3125b^5\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful way to expand expressions that are raised to a power. For any binomial expression \(x+y\)^n, the theorem states it can be expanded using a sum of terms: \( \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \).
This theorem essentially allows you to take a simple binomial like \(x+y\) and express it as a sum of multiple terms raised to different powers. Each term consists of a binomial coefficient, \(\binom{n}{k}\), along with the terms \(x^{n-k}\) and \(y^k\).
This is especially useful in algebra because it simplifies the process of multiplying a binomial by itself multiple times. In this exercise, the binomial theorem is applied to the expression \(4a - 5b\)^5, breaking it down into more manageable calculations.

Binomial Coefficient

The binomial coefficients are crucial in binomial expansions. Represented as \(\binom{n}{k}\), this notation is known as "n choose k". It defines the number of ways to pick \(k\) items out of \(n\) without considering the order.
In the context of binomial expansion, these coefficients determine the weight or size of each term in the expansion formula. The binomial coefficient is calculated using the formula:

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

Here, \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\). Each term in the binomial series \((4a - 5b)^5\) has its unique binomial coefficient, guiding the strength of each term's presence in the expansion.
As an example, for \(k = 2\), the binomial coefficient \(\binom{5}{2} = 10\) plays a pivotal role in the term \(4000a^3b^2\).

Polynomial Expansion

Expanding a binomial expression into a polynomial involves expressing it as a sum of terms, each containing powers of the binomial's components. A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, structured into multiple terms.
In our specific example \(4a - 5b\)^5, polynomial expansion means converting this into a series of terms involving \(a\) and \(b\).
The binomial theorem makes this process systematic, specifying each term's composition via powers and coefficients. By calculating for each \(k\) from 0 to 5, each term in the sequence is characterized by decreasing powers of \(a\) and increasing powers of \(b\).
This structured method helps in distributing the powers of a binomial, allowing for a clear solution through direct calculations rather than multiplying manually, ensuring accuracy and efficiency when tackling polynomials. Each resulting polynomial term, like \(1024a^5\) or \(-3125b^5\), forms part of the comprehensive expansion for \(4a - 5b\)^5.