Problem 30 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The expansion is \(729r^6 - 1458r^5s + 1215r^4s^2 - 540r^3s^3 + 135r^2s^4 - 18rs^5 + s^6\).

Step 1: Identify the Binomial Coefficients

For the binomial expression \((3r - s)^6\), identify the coefficients using the binomial theorem formula, which is given by: \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Here, \(a = 3r\), \(b = -s\), and \(n = 6\). The binomial coefficients are provided by the binomial coefficient \(\binom{n}{k}\), calculated as \(\frac{n!}{k!(n-k)!}\).

Step 2: Apply the Binomial Theorem

Expand the expression \((3r - s)^6\) using the terms of the binomial theorem. The general term of the expansion is given by \(\binom{6}{k} (3r)^{6-k} (-s)^k\), where \(k\) ranges from 0 to 6.

Step 3: Calculate Each Term of the Expansion

The expansion involves calculating each term by plugging in values of \(k = 0, 1, 2, 3, 4, 5, 6\):- For \(k=0\): \(\binom{6}{0} (3r)^6 (-s)^0 = 729r^6\)- For \(k=1\): \(\binom{6}{1} (3r)^5 (-s)^1 = -1458r^5s\)- For \(k=2\): \(\binom{6}{2} (3r)^4 (-s)^2 = 1215r^4s^2\)- For \(k=3\): \(\binom{6}{3} (3r)^3 (-s)^3 = -540r^3s^3\)- For \(k=4\): \(\binom{6}{4} (3r)^2 (-s)^4 = 135r^2s^4\)- For \(k=5\): \(\binom{6}{5} (3r)^1 (-s)^5 = -18rs^5\)- For \(k=6\): \(\binom{6}{6} (3r)^0 (-s)^6 = s^6\)

Step 4: Combine Terms of the Expansion

Add all the calculated terms together to form the complete expansion: \[729r^6 - 1458r^5s + 1215r^4s^2 - 540r^3s^3 + 135r^2s^4 - 18rs^5 + s^6\]

Key Concepts

Binomial Coefficients

Binomial coefficients are the heart of binomial expansions. They appear in each term of the expansion and are shown in expressions like \( \binom{n}{k} \), where \( n \) is the power to which the binomial is raised, and \( k \) is the specific term's index. Calculating the binomial coefficient uses the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
This formula basically tells us how many combinations of \( k \) elements can be selected from \( n \) elements. For example, in the problem \( (3r - s)^6 \), we calculate coefficients for each term with values of \( k \) ranging from 0 to 6.

• For \( k = 0 \), the binomial coefficient \( \binom{6}{0} \) is 1, meaning there is only one way to choose 0 elements from 6.
• For \( k = 3 \), \( \binom{6}{3} \) is 20, as there are 20 ways to choose 3 elements from 6.

These coefficients tell us the weight of each term in our binomial expansion.

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions that are raised to a power. It states that any binomial expression of the form \( (a + b)^n \) can be expanded as a sum of terms involving binomial coefficients. The theorem formula is: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \] In simple words, it tells us how to spread out the multiplication of terms in a binomial raised to a power.

For the expression \( (3r - s)^6 \), using the theorem, the terms are formulated with \( a = 3r \), \( b = -s \), and \( n = 6 \). Each term follows the pattern \( \binom{6}{k} (3r)^{6-k} (-s)^{k} \). This ensures correctly weighted contributions of the positive and negative parts of the expression.

Using this theorem significantly simplifies the process, replacing repeated multiplication with straightforward computation of binomial coefficients and powers.

Polynomial Expansion

Polynomial expansion refers to rewriting a polynomial expression in a more detailed form, showing each individual term separately. It's essentially breaking down the expression into a sum of multiple products.

When dealing with binomial expressions like \( (3r - s)^6 \), polynomial expansion involves identifying each term using the binomial theorem and calculating their coefficients and remaining terms.

• The first term in our example is \( 729r^6 \), representing the fully expanded form of \( (3r)^6 \).
• Another term, such as \(-1458r^5s\), comes from combining the coefficient \( \binom{6}{1} = 6 \) with powers \( (3r)^5 \) and \( (-s)^1 \).
• This pattern repeats across all possible values of \( k \), from 0 to 6.

By adding all terms together, you get the complete polynomial: \[ 729r^6 - 1458r^5s + 1215r^4s^2 - 540r^3s^3 + 135r^2s^4 - 18rs^5 + s^6 \].
Each term reflects aspects of the original expression, with its components clearly displayed and calculated.