Problem 45 - College Algebra with Corequisite Support

Question

In the expansion of \((5 x+3 y)^{n},\) each term has the form \(\left(\begin{array}{l}n \\ k\end{array}\right) a^{n-k} b^{k},\) where \(k\) successively takes on the value \(0,1,2, \ldots, n .\) If \(\left(\begin{array}{l}n \\ k\end{array}\right)=\left(\begin{array}{l}7 \\\ 2\end{array}\right),\) what is the corresponding term?

Step-by-Step Solution

Answer

The corresponding term is \(590625x^5y^2\).

Step 1: Understand Binomial Theorem

The expression \((5x + 3y)^n\) is expanded using the binomial theorem, which states that each term in the expansion can be written as \(\binom{n}{k} (5x)^{n-k} (3y)^k\). Here, \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify Given Binomial Coefficient

We are provided with \(\binom{n}{k} = \binom{7}{2}\). This means that the chosen value of \((n, k)\) relates to the given binomial coefficient where \(n = 7\) and \(k = 2\).

Step 3: Calculate the Required Term

Using \(n = 7\) and \(k = 2\), the corresponding term is given by the formula: \(\binom{7}{2} (5x)^{7-2} (3y)^2\).

Step 4: Calculate Binomial Coefficient

Compute \(\binom{7}{2}\) which is equal to \(\frac{7 \times 6}{2 \times 1} = 21\).

Step 5: Evaluate the Powers of Terms

Calculate \((5x)^{7-2} = (5x)^5\) and \((3y)^2\).

Step 6: Combine the Terms

Substitute all the calculations into the formula: \(21 \cdot (5x)^5 \cdot (3y)^2\).

Step 7: Simplify the Term

Simplify the expression: \((5x)^5 = 3125x^5\) and \((3y)^2 = 9y^2\). Therefore, the term is \(21 \cdot 3125x^5 \cdot 9y^2 = 590625x^5y^2\).

Key Concepts

Binomial Coefficient

The binomial coefficient is an important concept in mathematics, especially when dealing with expansions and combinatorics. It is represented as \( \binom{n}{k} \) and reads as "n choose k." The binomial coefficient calculates the number of ways to choose \( k \) elements from a set of \( n \) elements without considering the order of selection.

This concept is central to the Binomial Theorem, which states that any power of a binomial can be expanded into terms of coefficients and powers of the individual terms. The formula for the binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \( n! \) represents "n factorial," the product of all positive integers up to \( n \).
In our exercise, we calculated \( \binom{7}{2} \) as \( \frac{7 \times 6}{2 \times 1} = 21 \). This tells us how many ways we can pick 2 items from a set of 7. Understanding this is crucial not just for polynomial expansions but also in various fields like probability and statistics.

Polynomial Expansion

Polynomial expansion occurs when you expand an expression like \((a + b)^n\). The Binomial Theorem provides a systematic method to do this, yielding a polynomial consisting of \( n+1 \) terms. Each term of this expansion takes the form:

• \( \binom{n}{k} a^{n-k} b^k \)

Here, \( a \) and \( b \) are the parts of the original binomial expression, while \( k \) can range from 0 to \( n \).

Let's consider the exercise: the polynomial expansion of \((5x + 3y)^7\) uses this formula to determine each term with various powers of \(5x\) and \(3y\), depending on the value of \( k \).
For instance, when \( k=2 \), we have the term \( \binom{7}{2} (5x)^{7-2}(3y)^2 \), leading to a coefficient of 21 multiplied by specific powers of \(5x\) and \(3y\). This not only simplifies computations but also allows us to predict patterns in polynomial arrangements.

Combinatorics

Combinatorics is a fascinating branch of mathematics dealing with counting, arrangement, and combination of objects. It is deeply interconnected with concepts like binomial coefficients that are fundamental in polynomial expansions.
The idea of choosing \( k \) elements from \( n \) without regard to order translates directly into combinatorial problems.

• For example, using \( \binom{n}{k} \), you determine how many ways you can select \( k \) items from a group of \( n \).

In the specific context of the Binomial Theorem, this technique helps identify the coefficients of terms in a binomial expansion.

Combinatorial principles apply in real-world scenarios too, like calculating probabilities or even optimizing paths in algorithms. Understanding these principles empowers students to solve complex problems by breaking them into simpler, countable parts. The exercise highlights this with its focus on selecting which powers to multiply and combine, a process made orderly and systematic through combinatorics.