Problem 21 - Algebra 2

Question

Find the indicated term of each expansion. sixth term of \((x-y)^{9}\)

Step-by-Step Solution

Answer

The sixth term is \(-126x^4y^5\).

Step 1: Understand the Problem Statement

We are asked to find the sixth term in the expansion of \((x-y)^9\). This requires using the Binomial Theorem, which provides a method to expand expressions of the form \((a + b)^n\).

Step 2: Recall the Binomial Theorem

The Binomial Theorem states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = x\), \(b = -y\), and \(n = 9\).

Step 3: Identify the Desired Term

The \((k+1)\)-th term in the expansion is given by \(\binom{n}{k} a^{n-k} b^k\). For the sixth term, we need the \((5+1)\)-th term, where \(k = 5\).

Step 4: Substitute Values in the Expression

Substitute \(n = 9\), \(a = x\), \(b = -y\), and \(k = 5\) into the binomial expression. The sixth term will be: \(\binom{9}{5} x^{9-5} (-y)^5\).

Step 5: Calculate the Binomial Coefficient

Calculate \(\binom{9}{5}\), which is the same as \(\binom{9}{4}\). Use the formula for binomial coefficients: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). We have: \(\binom{9}{5} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126\).

Step 6: Compute the Powers

Compute \(x^{9-5} = x^4\) and \((-y)^5 = -y^5\).

Step 7: Assemble the Term

Combine the computed values to assemble the sixth term: \(126 \times x^4 \times (-y)^5 = 126x^4(-y)^5\). Simplify it to \(-126x^4y^5\).

Step 8: Simplify the Sign

The negative sign comes from \((-y)^5\), which is negative because the exponent 5 is odd. Thus, the sign of the term is negative, leading to \(-126x^4y^5\).

Key Concepts

Binomial Expansion

The Binomial Theorem is a fundamental concept in algebra that allows us to expand expressions raised to a power, such as \( (x-y)^{9} \). It provides a formula for expressing \( (a + b)^{n} \) as a sum of terms involving powers of \( a \) and \( b \), known as the binomial expansion.
This formula is expressed as:

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

Here, each term in the expansion is determined by a specific binomial coefficient. Each term represents the different ways of selecting elements from the binomial. This method simplifies the process of expanding higher degree polynomial expressions efficiently.
In our practice exercise, the Binomial Theorem helps us identify a specific term in the polynomial expansion of \( (x-y)^{9} \), ensuring precision and correctness by methodically calculating terms without manually expanding the entire polynomial.

Polynomial Expansion

Polynomial Expansion is the process of expressing a polynomial raised to a power as a sum of its individual terms. This process is crucial because it helps in solving complex polynomial equations and expressions.
For a polynomial like \( (x-y)^9\), expanding it term by term using the Binomial Theorem allows us to unravel it into simpler terms like \( x^9 \), \( -9x^8y \), etc., \( -126x^4y^5 \) representing the 6th term as derived from the binomial expression.
This technique not only simplifies calculations but also aids in analyzing polynomial functions, as you're able to directly work with each individual term. Furthermore, polynomial expansions help in visualizing how changes in variables affect the function as a whole, which is particularly useful in calculus and higher-level mathematics.

Binomial Coefficient

The Binomial Coefficient is a crucial element of the Binomial Theorem. It indicates the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order. The Binomial Coefficient is represented as \( \binom{n}{k} \).
It is calculated using the formula:

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

In our example problem, we found \( \binom{9}{5} \), which means choosing 5 elements from 9. Calculating \( \binom{9}{5} \) gives us 126, which is a crucial step in determining the specific term in the expansion of the binomial.
This coefficient helps determine the magnitude of each term in the expansion, providing insight into the polynomial’s distribution of terms and their respective powers. Recognizing and calculating the Binomial Coefficient correctly is essential for successful polynomial and binomial manipulations.