Problem 22 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(a + 5\right)^5 \)

Step-by-Step Solution

Answer

The simplified form of the expression \( \left(a + 5\right)^5 \) is \( a^5 + 25a^4 + 250a^3 + 1250a^2 + 3125a + 3125 \).

Step 1: Recognize the Binomial Theorem pattern

The Binomial theorem states: \( (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \). In this case, \( a = a \), \( b = 5 \), and \( n = 5 \).

Step 2: Apply Binomial Theorem

Applying the binomial theorem to \( \left(a + 5\right)^5 \) results in \( \left(a + 5\right)^5 = \sum_{k=0}^{5} {5 \choose k} a^{5-k} 5^k \). This expands to \( 5^0a^{5-0}{5 \choose 0} + 5^1a^{5-1}{5 \choose 1} + 5^2a^{5-2}{5 \choose 2} + 5^3a^{5-3}{5 \choose 3} + 5^4a^{5-4}{5 \choose 4} + 5^5a^{5-5}{5 \choose 5} \).

Step 3: Simplify the expression

Now we just need to calculate the choose operation and simplify. The expression becomes: \( a^5 + 5*5a^4 + 10*25a^3 + 10*125a^2 + 5*625a + 3125 \).

Step 4: Final answer

Simplifying further we get: \( a^5 + 25a^4 + 250a^3 + 1250a^2 + 3125a + 3125 \).

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. This allows for easier calculations and understanding of the underlying structure of the polynomial. In the context of the Binomial Theorem, polynomial expansion helps break down expressions like \( (a + b)^n \) into simpler parts.
\( (a + b)^5 \), for example, can be expanded to a sum of terms where each term consists of products between powers of \(a\) and \(b\). This is achieved by systematically changing the powers starting from \(a^5\) and going to \(b^5\).

• Each product and sum in polynomial expansions follows a strict pattern dictated by the powers of the terms.
• Understanding how to expand a polynomial is fundamental, as it simplifies algebraic expressions and makes them more manageable.

Once fully expanded using the Binomial Theorem, each term in a polynomial expansion can be interpreted through its coefficient, its power of \(a\), and its power of \(b\). This structured approach aids in performing more complex mathematical operations efficiently.

Combinatorial Coefficients

Combinatorial coefficients, often shown as \( {n \choose k} \), are crucial when working with polynomial expansions, particularly in the application of the Binomial Theorem. These coefficients determine how many ways we can choose \(k\) elements from \(n\) elements, and they weigh each term in the expansion of \((a + b)^n\).
The formula for the binomial coefficient is:
\[ {n \choose k} = \frac{n!}{k!(n-k)!} \]
where \(!\) denotes factorial, which is the product of all positive integers up to that number.

• In our example \((a + 5)^5\), the coefficients for each term are found by plugging \(n=5\) and \(k\) values ranging from 0 to 5 into the formula.
• These coefficients are what give the expansion its distinct structure.

The resulting coefficients \([1, 5, 10, 10, 5, 1]\) help to build terms like \(5^k\) in combination with the powers of \(a\) and \(b\), creating balance and symmetry in the final polynomial.

Pascal's Triangle

Pascal's Triangle is a practical visualization that helps in finding combinatorial coefficients for polynomial expansions. Each row of Pascal's Triangle corresponds to the coefficients in the expansion of \((a + b)^n\).
Created in a simple, iterative fashion, it starts with a \(1\) at the top, each number is the sum of the two directly above it. This triangular representation makes it easy to reference coefficients quickly.

• The \(n\)th row of Pascal's Triangle aligns with the coefficients for \((a + b)^n\).
• For \((a + 5)^5\), the fifth row \([1, 5, 10, 10, 5, 1]\) provides the necessary coefficients for expansion.

By using Pascal's Triangle, you can easily find the combinatorial numbers needed for polynomial expansions, especially in the context of the Binomial Theorem. This technique streamlines the process, allowing learners to focus on understanding how the theorem orchestrates polynomial transformations.