Problem 34 - College Algebra

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x-2 y)^{9} $$

Step-by-Step Solution

Answer

The first three terms in the binomial expansion of \( (x-2y)^9 \) in simplified form are \(x^9, -18x^8y,\) and \( 144x^7y^2\).

Step 1: Apply the binomial theorem

Apply the binomial theorem to obtain the first three terms. Mathematically, \( (a+b)^n = {n\choose 0} a^n b^0 + {n\choose 1} a^{n-1} b^1 + {n\choose 2} a^{n-2} b^2 + ...\), we can use this formula to write the terms. Here our \( a=x \), \( b=-2y \) and \( n=9 \).

Step 2: Compute the individual terms

We can use binomial coefficients and substitute the values into the formula. The first three terms will be: \({9\choose 0} x^9 (-2y)^0 = 1 \cdot x^9 \cdot 1 = x^9\), \({9\choose 1} x^8 (-2y)^1 = 9 \cdot x^8 \cdot (-2y) = -18x^8y\), \({9\choose 2} x^7 (-2y)^2 = 36 \cdot x^7 \cdot 4y^2 = 144x^7y^2\).

Step 3: Write the first three terms

After computing the individual terms, the first three terms of the binomial expansion are: \(x^9, -18x^8y, 144x^7y^2\).

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that allows us to expand expressions of the form \((a + b)^n\) into a sum of terms involving binomial coefficients. This theorem is essential for simplifying and expanding expressions raised to a power. It provides a systematic way to calculate each term in the expansion without manually multiplying everything out.

To apply the binomial theorem, consider the formula:

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

This formula means that each term in the expansion consists of:

• A binomial coefficient \({n \choose k}\)
• \(a\) raised to the power of \((n-k)\)
• \(b\) raised to the power of \(k\)

By understanding this pattern, you can systematically find each term in a binomial expansion.

Binomial Coefficients

Binomial coefficients are the numbers that appear in the expansion of a binomial expression. These coefficients are crucial when using the binomial theorem, as they determine the weight of each term in the expanded expression.

Mathematically, binomial coefficients are represented as \({n \choose k}\), and this represents the number of ways to choose \(k\) elements from a set of \(n\) elements. The formula for binomial coefficients is:

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

Where \(n!\) (n factorial) is the product of all positive integers up to \(n\). Each coefficient determines how the powers of \(a\) and \(b\) combine in a given term. For example, in the binomial expansion of \((x - 2y)^9\), the binomial coefficients for the first three terms are \(1\), \(9\), and \(36\). These coefficients govern how the variables are combined and expanded in each step of the process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). These expressions can represent real-world quantities or purely mathematical ideas. Understanding how to manipulate and expand algebraic expressions is vital in solving problems across various areas in mathematics.

When dealing with binomial expansions, an expression like \((x - 2y)^9\) is broken down into simpler parts using both the binomial theorem and binomial coefficients. Here, the terms in the expression are derived from expanding \((a + b)^n\) into a series of algebraic expressions that include different powers of \(x\) and \(y\). Once expanded, each term can be further simplified by performing the necessary arithmetic operations, combining like terms, or converting expressions into more recognizable forms.

These expressions help represent complex relationships in an understandable way, forming the backbone of problem solving in algebra.