Problem 30 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$(3 r-s)^{6}$$

Step-by-Step Solution

Answer

The expansion is \(729r^6 - 1458r^5s + 1215r^4s^2 - 1620r^3s^3 + 405r^2s^4 - 18rs^5 + s^6\).

Step 1: Understand the Binomial Theorem

The binomial theorem is used to expand expressions of the form \((a + b)^n\). It is given by the formula \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] where \(\binom{n}{k}\) is a binomial coefficient.

Step 2: Identify the Terms

In this problem, the expression is \((3r-s)^6\). Here, \(a = 3r\), \(b = -s\), and \(n = 6\). We will use these to apply the binomial theorem.

Step 3: Calculate Binomial Coefficients

The binomial coefficients are given by \(\binom{n}{k}\), which is equal to \(\frac{n!}{k!(n-k)!}\). For \(n = 6\), the coefficients from \(k=0\) to \(k=6\) are: \(1, 6, 15, 20, 15, 6, 1\).

Step 4: Expand Using the Binomial Formula

Substitute the terms \(a = 3r\) and \(b = -s\) into the binomial expansion formula:\[(3r-s)^6 = \sum_{k=0}^{6} \binom{6}{k} (3r)^{6-k} (-s)^k\]Calculate each term from \(k=0\) to \(k=6\):

Step 5: Substitute and Simplify Each Term

Expand each term:- For \(k = 0\): \(\binom{6}{0}(3r)^6(-s)^0 = 1 \times (3r)^6 \times 1 = 729r^6\)- For \(k = 1\): \(\binom{6}{1}(3r)^5(-s)^1 = 6 \times 243r^5 \times (-s) = -1458r^5s\)- For \(k = 2\): \(\binom{6}{2}(3r)^4(-s)^2 = 15 \times 81r^4s^2 = 1215r^4s^2\)- For \(k = 3\): \(\binom{6}{3}(3r)^3(-s)^3 = 20 \times 27r^3(-s)^3 = -1620r^3s^3\)- For \(k = 4\): \(\binom{6}{4}(3r)^2(-s)^4 = 15 \times 9r^2s^4 = 405r^2s^4\)- For \(k = 5\): \(\binom{6}{5}(3r)^1(-s)^5 = 6 \times 3r(-s)^5 = -18rs^5\)- For \(k = 6\): \(\binom{6}{6}(3r)^0(-s)^6 = 1 \times 1 \times s^6 = s^6\)

Step 6: Write the Expanded Expression

Combine all the terms together to form the expanded expression: \[729r^6 - 1458r^5s + 1215r^4s^2 - 1620r^3s^3 + 405r^2s^4 - 18rs^5 + s^6\]

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power, such as \((a + b)^n\). This theorem is essential because it simplifies the process of expanding and calculating polynomial sums.
The formula is given by: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]

• \((a + b)^n\): Represents the binomial expression where \(a\) and \(b\) are any two terms and \(n\) is a non-negative integer.
• \(\binom{n}{k}\): These are the binomial coefficients, representing the magnitude of each term in the expansion.
• The symbol \(\sum\) indicates summation, meaning we are adding all the terms from \(k = 0\) to \(n\).

Using the binomial theorem not only saves time in calculations but also helps in deriving further algebraic theories. It provides a structured method to calculate terms systematically.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operations. In our binomial expansion problem, the expression \((3r - s)^6\) is an example.
Algebraic expressions can differ based on terms and can be manipulated to simplify or solve problems. Here:

• Numerical Coefficients: Numbers that appear in front of the variables, like 3 in \(3r\).
• Variables: Symbols that represent numbers or quantities, such as \(r\) and \(s\).
• Operations: Includes arithmetic operations like addition, subtraction, multiplication, and division.

Algebraic expressions are fundamental in mathematics as they form the basis for constructing equations and understanding more complex structures. Learning how to manipulate these expressions efficiently is crucial in mathematics and other related fields.

Binomial Coefficient

The binomial coefficient is a key component in the binomial expansion. These coefficients count the number of combinations or ways to choose \(k\) elements from \(n\) elements without considering the order and are represented by \(\binom{n}{k}\).
The formula to calculate these coefficients is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• \(n!\) (n factorial): The product of all positive integers up to \(n\). For example, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
• \(k!\) and \((n-k)!\): Represent the product of positive integers up to \(k\) and \(n-k\), respectively.

These binomial coefficients are crucial as they determine the magnitude of each term in the polynomial expansion and facilitate calculations in combinatorics and algebra.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a binomial raised to a power as a sum of terms. In this case, \((3r-s)^{6}\) is expanded using the binomial theorem.

• Each term in the expansion has a binomial coefficient that determines its size.
• The powers of \(3r\) and \(-s\) decrease and increase in tandem as the terms are expanded, starting from \((3r)^6\) and \((-s)^0\) to \((3r)^0\) and \((-s)^6\).

By calculating and combining each term, the original compact expression is now a comprehensive polynomial:\[729r^6 - 1458r^5s + 1215r^4s^2 - 1620r^3s^3 + 405r^2s^4 - 18rs^5 + s^6\]This transformation simplifies further analysis, allowing for calculations such as evaluation and simplification.