Problem 32 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\((4a - 5b)^5 = 1024a^5 - 6400a^4b + 16000a^3b^2 - 20000a^2b^3 + 12500ab^4 - 3125b^5\).

Step 1: Identify the binomial expression

The given expression is \((4a - 5b)^5\), which is a binomial expression of the form \((x + y)^n\). Here, \(x = 4a\), \(y = -5b\), and \(n = 5\).

Step 2: Binomial Theorem formula

The binomial theorem states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). This formula will guide us in expanding \((4a - 5b)^5\).

Step 3: Calculate binomial coefficients

Use the binomial coefficient formula \(\binom{n}{k}=\frac{n!}{k!(n-k)!}\) to find the coefficients from \(\binom{5}{0}\) to \(\binom{5}{5}\). The coefficients are: 1, 5, 10, 10, 5, 1.

Step 4: Substitute and simplify terms

For each term, substitute \(x = 4a\), \(y = -5b\), and the binomial coefficient \(\binom{5}{k}\) into \((x+y)^n\). Calculate individual terms: - Term 1: \(\binom{5}{0}(4a)^5(-5b)^0 = 1 \cdot 1024a^5 \cdot 1 = 1024a^5\) - Term 2: \(\binom{5}{1}(4a)^4(-5b)^1 = 5 \cdot 256a^4 \cdot (-5b) = -6400a^4b\) - Term 3: \(\binom{5}{2}(4a)^3(-5b)^2 = 10 \cdot 64a^3 \cdot 25b^2 = 16000a^3b^2\) - Term 4: \(\binom{5}{3}(4a)^2(-5b)^3 = 10 \cdot 16a^2 \cdot (-125b^3) = -20000a^2b^3\) - Term 5: \(\binom{5}{4}(4a)^1(-5b)^4 = 5 \cdot 4a \cdot 625b^4 = 12500ab^4\) - Term 6: \(\binom{5}{5}(4a)^0(-5b)^5 = 1 \cdot 1 \cdot (-3125b^5) = -3125b^5\)

Step 5: Write the final expanded form

Combine all the terms to obtain the full binomial expansion of \((4a - 5b)^5\): \[1024a^5 - 6400a^4b + 16000a^3b^2 - 20000a^2b^3 + 12500ab^4 - 3125b^5\].

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful formula in algebra that provides a way to expand expressions of the form \((x + y)^n\). It gives us a systematic method to expand any binomial raised to a power. According to the theorem, you can express the expansion as a sum of terms given by:\[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]Each term in the expansion has a specific structure:

• \(x\) and \(y\) are the individual terms of the binomial.
• \((n-k)\) and \(k\) denote the powers of \(x\) and \(y\), respectively.
• \(\binom{n}{k}\) represents the binomial coefficient.

The Binomial Theorem simplifies solving expressions vastly, reducing the manual computation needed for complex powers.

Binomial Coefficients

Binomial coefficients are a fundamental component of the Binomial Theorem. They are used to calculate the number of ways to choose \(k\) items from \(n\) items, symbolized as \(\binom{n}{k}\), read as "n choose k".The formula for finding a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This formula indicates that:

• \(n!\) is the factorial of \(n\).
• \(k!\) is the factorial of \(k\).
• \((n-k)!\) is the factorial of \((n-k)\).

Understanding binomial coefficients is crucial for accurately expanding binomials using the theorem as they determine the weight each term in the expansion holds.

Factorial

In mathematics, factorials are used to determine the product of an integer and all the positive integers below it. The notation for factorial is the exclamation point \(!\). For example, \(5!\) (read as "five factorial") is:\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]Factorials are especially significant in the calculation of binomial coefficients. They provide the basis for determining the number of permutations and combinations, which is central to probability and combinatorics.By understanding factorials, you can easily get the binomial coefficients needed for expanding expressions using the Binomial Theorem.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In the context of binomials, expressions such as \((4a - 5b)^5\) showcase the use of algebraic components.Key points about algebraic expressions include:

• Variables: These are symbols that represent values that can change, like \(a\) and \(b\) in our example.
• Constants: These are fixed numbers within the expression, like the numbers 4 and -5.
• Operations: Determine the relationships between numbers and variables, crucially affecting how expressions are expanded or solved.

Algebraic expressions form the building blocks of more complex mathematical concepts, enabling us to model real-world situations and perform various types of mathematical analysis.