Problem 33 - College Algebra

Question

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x-2 y)^{10}$$

Step-by-Step Solution

Answer

The first three terms in the binomial expansion of \( (x-2 y)^{10}\) are: \(x^{10}, -20x^9y, 180x^8y^2\).

Step 1: Identify the parameters

In this case, we have \(a = x\), \(b = (-2y)\) and \(n = 10\). According to binomial theorem, the first three terms are given by: \(C(n, 0)a^n b^0, C(n, 1) a^{n-1}b^1, C(n, 2) a^{n-2}b^2\).

Step 2: Calculate the coefficients

The binomial coefficients \(C(n, 0), C(n, 1)\) and \(C(n, 2)\) are calculated as: \(C(10, 0) = 1\), \(C(10, 1) = 10\), \(C(10, 2) = 45\).

Step 3: Calculate the first term

The first term of the expansion is given by the formula \(C(n, 0)a^n b^0\). So its value equals \(1.x^{10}.(-2y)^{0} = x^{10}\).

Step 4: Calculate the second term

The second term of the expansion is given by the formula \(C(n, 1) a^{n-1}b^1\). So its value equals \(10.x^{9}.(-2y)^1 = -20x^9y\).

Step 5: Calculate the third term

The third term of the expansion is given by the formula \(C(n, 2) a^{n-2}b^2\). So its value equals \(45.x^{8}.(-2y)^2 = 180x^8y^2\).

Key Concepts

Binomial Expansion

The Binomial Expansion is a fundamental concept in algebra that allows us to expand expressions raised to a power, typically shown in the form \((a + b)^n\). In our given exercise, the expression is \((x - 2y)^{10}\). Binomial expansion provides a formula to express the powers of binomials in expanded form, using the binomial theorem. The theorem is structured as follows:

• Identify the value of \(a\), \(b\), and \(n\). In this case, \(a = x\), \(b = -2y\), and \(n = 10\).
• Use the binomial coefficients \(C(n, k)\) for each term.
• Apply the general term formula \(C(n, k) a^{n-k} b^k\) to find each term in the expansion.

This theorem simplifies the task of expanding large powers neatly into a series of terms.

Coefficients

Coefficients are crucial in the expansion process, as they determine the multiplicative factor for each term in the binomial expansion. In the context of our problem, coefficients are computed using the binomial coefficient \(C(n, k)\), also known as "n choose k." This represents the number of combinations of \(n\) items taken \(k\) at a time:

• For the first term: \(C(10, 0) = 1\).
• For the second term: \(C(10, 1) = 10\).
• For the third term: \(C(10, 2) = 45\).

These coefficients are essential as they scale the terms in a binomial expansion, determining how the numeric part of each expanded term changes with respect to its position in the sequence.

Algebraic Expression

An Algebraic Expression is a mathematical phrase that can include numbers, variables, and operation symbols. In binomial expansion, we deal with expressions such as \((x - 2y)^{10}\), where the expansion results in a series of algebraic expressions. Each term of this expression is influenced by the coefficients derived from the binomial theorem:

• The first term is \(x^{10}\), a straightforward expression with only a single variable in a high power.
• The second term, \(-20x^9y\), introduces both variables and a negative sign, showing how the binomial can modify variables based on their position.
• The third term, \(180x^8y^2\), increases the complexity by raising the power further, including a change in coefficient size.

Through such expressions, you can assess the significance of algebraic operations and how they interact at different steps in a mathematical sequence. Whether evaluating or manipulating these expressions, understanding their foundational structure aids in both solving complex problems and simplifying results.