Problem 27 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$\left(r^{2}+s\right)^{5}$$

Step-by-Step Solution

Answer

\((r^2+s)^5 = r^{10} + 5r^8s + 10r^6s^2 + 10r^4s^3 + 5r^2s^4 + s^5\)

Step 1: Identify the Binomial Theorem

To expand the expression \((r^2+s)^5\), we'll use the Binomial Theorem, which states that \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \(a = r^2\), \(b = s\), and \(n = 5\).

Step 2: Write the General Term of the Expansion

The general term in the binomial expansion \(T_k\) can be written as \(\binom{5}{k} (r^2)^{5-k} s^k\). This formula will help us write each term of the expansion from \(k=0\) to \(k=5\).

Step 3: Calculate Each Term of the Expansion

We calculate each term for \(k = 0\) to \(k = 5\): - **For \(k=0\):** \(T_0 = \binom{5}{0} (r^2)^5 s^0 = 1 \cdot r^{10} \cdot 1 = r^{10}\)- **For \(k=1\):** \(T_1 = \binom{5}{1} (r^2)^4 s^1 = 5 \cdot r^8 \cdot s = 5r^8s\)- **For \(k=2\):** \(T_2 = \binom{5}{2} (r^2)^3 s^2 = 10 \cdot r^6 \cdot s^2 = 10r^6s^2\)- **For \(k=3\):** \(T_3 = \binom{5}{3} (r^2)^2 s^3 = 10 \cdot r^4 \cdot s^3 = 10r^4s^3\)- **For \(k=4\):** \(T_4 = \binom{5}{4} (r^2)^1 s^4 = 5 \cdot r^2 \cdot s^4 = 5r^2s^4\)- **For \(k=5\):** \(T_5 = \binom{5}{5} (r^2)^0 s^5 = 1 \cdot 1 \cdot s^5 = s^5\)

Step 4: Combine the Terms

Combine all the terms from \(k=0\) to \(k=5\) to write the full binomial expansion:\[ (r^2+s)^5 = r^{10} + 5r^8s + 10r^6s^2 + 10r^4s^3 + 5r^2s^4 + s^5 \]

Key Concepts

Binomial Expansion

The Binomial Theorem is a powerful tool for expanding expressions raised to a power, like \((r^2+s)^5\). This theorem tells us how to break down the expansion into individual terms. The formula for the Binomial Theorem is:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

In this case:

• \(a = r^2\)
• \(b = s\)
• \(n = 5\)

By applying these values, we can expand the expression step by step. Each term in the expansion is calculated using the combination formula \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from \(n\) elements. This step-by-step process allows for precise calculation of each term, resulting in the complete expansion: \[ (r^2+s)^5 = r^{10} + 5r^8s + 10r^6s^2 + 10r^4s^3 + 5r^2s^4 + s^5 \] By understanding each part of the binomial expansion, you can tackle similar problems with confidence.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. It's essential for calculating the binomial coefficients in the Binomial Theorem. In this exercise, we used combinatorics to find the coefficients for each term in the expansion of \((r^2+s)^5\).
Here's how it works:

• \(\binom{5}{k}\) is the binomial coefficient, which tells us how many ways we can choose \(k\) elements from 5.
• The coefficient is calculated as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

For example:

• \(\binom{5}{0} = 1\)
• \(\binom{5}{1} = 5\)
• \(\binom{5}{2} = 10\)

These coefficients are crucial, as they directly determine the size of each term in the binomial expansion. By mastering these calculations, you enhance your problem-solving skills in algebra and beyond.

Polynomial Expressions

Polynomial expressions, like \((r^2+s)^5\), are mathematical phrases involving sums of power terms. Expanding them using the Binomial Theorem converts the expression into a series of polynomial terms.
Here's what makes polynomials so interesting:

• They consist of variables with whole-number exponents.
• Each term is a product of a coefficient and a power of the variable(s).
• The degree of the polynomial is the highest power found in the expression.

In our example, \((r^2+s)^5\) is expanded into several terms with different powers of \(r^2\) and \(s\). Each of these terms, like \(r^{10}\) or \(5r^8s\), fits into the structure of a polynomial, showing the flexibility and utility of polynomials in expressing complex relationships in algebra. Understanding polynomials and how they expand is key to solving many algebraic problems effectively. By breaking down the expression into its polynomial components, we make the math more approachable and less intimidating.