Problem 29 - Precalculus Mathematics for Calculus

Question

Find the first three terms in the expansion of \((x+2 y)^{20}\).

Step-by-Step Solution

Answer

The first three terms are \(x^{20}\), \(40x^{19}y\), and \(760x^{18}y^2\).

Step 1: Recall the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is a binomial coefficient. For our problem, substitute \(a = x\), \(b = 2y\), and \(n = 20\).

Step 2: Calculate the First Term

The first term corresponds to \(k=0\). Using the formula: \(\binom{20}{0}x^{20}(2y)^0 = 1 \cdot x^{20} \cdot 1 = x^{20}\). So, the first term is \(x^{20}\).

Step 3: Calculate the Second Term

The second term corresponds to \(k=1\). Substitute into the formula: \(\binom{20}{1}x^{19}(2y)^1 = 20 \cdot x^{19} \cdot 2y = 40x^{19}y\). Thus, the second term is \(40x^{19}y\).

Step 4: Calculate the Third Term

The third term corresponds to \(k=2\). Substitute into the formula: \(\binom{20}{2}x^{18}(2y)^2 = 190 \cdot x^{18} \cdot 4y^2 = 760x^{18}y^2\). Therefore, the third term is \(760x^{18}y^2\).

Step 5: List First Three Terms

The first three terms of the expansion \((x+2y)^{20}\) are \(x^{20}, 40x^{19}y, \) and \(760x^{18}y^2\).

Key Concepts

Binomial Coefficient

In the context of the binomial expansion, the binomial coefficient is an essential component. It represents the number of ways to choose a subset of items from a larger set, which in mathematical terms, is expressed as:

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

Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \). The binomial coefficient \( \binom{n}{k} \) is also known as "n choose k."
Understanding this concept is crucial because it quantifies how many different ways we can select \( k \) elements from \( n \) elements. In a binomial expansion like \((x + 2y)^{20}\), calculating the coefficients for each term involves determining these choices.
This is why you'll often see binomial coefficients prominently in polynomial expansions using the Binomial Theorem. For example, in Step 4, we calculated \( \binom{20}{2} = 190 \) as part of finding the third term in the expression's expansion.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power as a sum of terms. Each term includes a coefficient, derived from the binomial coefficient, and variables raised to various powers.
The polynomial expansion of a binomial form \((a + b)^n\) is specifically addressed through the Binomial Theorem:

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

Each element in the sum is a result of expanding the expression \((a + b)\) multiplied by itself \( n \) times.
This results in several terms where powers of \( a \) and \( b \) decrease by one for \( a \) and increase by one for \( b \), respectively. The binomial coefficient \( \binom{n}{k} \) helps determine the exact multiplier for each term.

Binomial Expansion

Binomial expansion is a method used to expand expressions of the form \((a + b)^n\). It is a specific application of polynomial expansion where only two terms are involved: one known as \( a \) and the other as \( b \). The Binomial Theorem, which we've used earlier, guides this expansion:

• \((x + 2y)^{20} = \sum_{k=0}^{20} \binom{20}{k} x^{20-k} (2y)^k\)

In essence, this tells us how \((x + 2y)\) magnifies or expands when raised to the 20th power.
Each step in the expansion requires careful application of the binomial coefficient and powers of \( x \) and \( 2y \). The first term \( x^{20} \) stems from zero usage of \( 2y \) (i.e., \( k=0 \)), while the third term \( 760x^{18}y^2 \) results from \( k=2 \), where two components of \( 2y \) are included.
Binomial expansion gives us a practical method to approach large polynomial expressions systematically, ensuring no term is missed.