Problem 27 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(2x + y\right)^3 \)

Step-by-Step Solution

Answer

The expansion and simplification of \( (2x + y)^3 \) using the Binomial Theorem is \( 8x^3 + 12x^2*y + 6x*y^2 + y^3 \)

Step 1: Identify the binomial expression

The binomial expression in the problem is \( (2x + y)^3 \). Here, a=2x, b=y, and n=3.

Step 2: Expand using the Binomial Theorem

Using the Binomial Theorem, \( (2x + y)^3 \) expands to \( C(3, 0) * (2x)^3 * y^0 + C(3, 1) * (2x)^2 * y^1 + C(3, 2) * (2x)^1 * y^2 + C(3, 3) * (2x)^0 * y^3 \).

Step 3: Calculate binomial coefficients and simplify terms

We have to calculate the binomial coefficients C(3, k) for k=0 to 3. Using the formula for C(n, k) gives C(3, 0) = 1, C(3, 1) = 3, C(3, 2) = 3, C(3, 3) = 1. So, the expanded expression becomes: \( 1*(2x)^3 * y^0 + 3*(2x)^2 * y^1 + 3*(2x)^1 * y^2 + 1*(2x)^0 * y^3 = 8x^3 + 12x^2*y + 6x*y^2 + y^3 \).

Key Concepts

Binomial Coefficients

The concept of binomial coefficients plays a crucial role in expanding binomials using the Binomial Theorem. Binomial coefficients are represented as \( C(n, k) \), which is often read as "n choose k". These coefficients give us the number of ways to choose \( k \) elements from \( n \) elements without considering the order. In mathematical terms, it is calculated using the formula:
\[ C(n, k) = \frac{n!}{k! (n-k)!} \]
Here's what you need to know:

• \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \).
• Each term in a binomial expansion corresponds to a binomial coefficient.
• The sum of the coefficients for any row in Pascal's Triangle (used to find binomial coefficients) is \( 2^n \).

In our example \((2x + y)^3\), the binomial coefficients are \( C(3, 0) = 1 \), \( C(3, 1) = 3 \), \( C(3, 2) = 3 \), and \( C(3, 3) = 1 \). Calculating these coefficients allows us to determine the weights of each term in the expanded polynomial expression.

Expansion of Expressions

When expanding expressions such as \((2x + y)^3\) using the Binomial Theorem, the goal is to express the binomial as a sum of terms raised to various powers. The Binomial Theorem provides us with a clear and systematic way to do this:
For a binomial \((a + b)^n\), the expansion is given by:
\[ \sum_{k=0}^{n} C(n, k) \cdot a^{n-k} \cdot b^{k} \]
Here's how the expansion process works:

• Each term in the expanded expression includes a binomial coefficient (\( C(n, k) \)).
• The powers of \( a \) decrease from \( n \) to 0, while the powers of \( b \) increase from 0 to \( n \).
• For our binomial \((2x + y)^3\), we calculate each term: \( (2x)^3, (2x)^2 \cdot y, (2x)^1 \cdot y^2, y^3 \).

These terms combine with their respective binomial coefficients to form the complete expansion.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A polynomial's degree is defined by the highest power of the variable present in the expression.
When expanding binomials like \((2x + y)^3\), the resulting expression is a polynomial:

• It contains several terms where each term is a product of constants and variables raised to an exponent.
• The polynomial resulting from our example is \( 8x^3 + 12x^2y + 6xy^2 + y^3 \).
• The polynomial has a degree of 3 since the highest power of any variable term is \( x^3 \).

Polynomials are fundamental in algebra as they form the basis for more complex equations and functions. Understanding how to manipulate and expand them is vital for solving algebraic problems efficiently.