Problem 29 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Binomial Theorem

To find the expansion of \((x + 2y)^{20}\), we will use the Binomial Theorem, which states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This formula helps us identify each term of the expansion.

Step 2: Compute the First Term (k=0)

For the first term, use \(k=0\) in the binomial expression: \[\binom{20}{0} x^{20-0} (2y)^0 = 1 \times x^{20} \times 1 = x^{20}.\]

Step 3: Compute the Second Term (k=1)

For the second term, use \(k=1\): \[\binom{20}{1} x^{20-1} (2y)^1 = 20 \times x^{19} \times 2y = 40x^{19}y.\]

Step 4: Compute the Third Term (k=2)

For the third term, use \(k=2\): \[\binom{20}{2} x^{20-2} (2y)^2 = 190 \times x^{18} \times 4y^2 = 760x^{18}y^2.\]

Step 5: Compile the First Three Terms

Combine the three terms we've calculated: \[x^{20}, 40x^{19}y, 760x^{18}y^2.\] These represent the first three terms of the expansion.

Key Concepts

Binomial Expansion

The Binomial Expansion is a crucial concept in algebra that allows us to expand expressions of the form \((a + b)^n\). The Binomial Theorem provides a formula, so we do not need to multiply everything out manually. This makes computations much more manageable, especially for large exponents. The theorem states: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this formula, \(\binom{n}{k}\) is a binomial coefficient, which defines the number of ways to choose \(k\) items from \(n\) without regard to the order. The terms \(a^{n-k}\) and \(b^k\) give us each term's contribution from \(a\) and \(b\) respectively for a given \(k\). Understanding how to apply the binomial theorem is fundamental for expanding binomials and obtaining specific terms. For example, in expanding \((x + 2y)^{20}\), we simply identify \(a = x\), \(b = 2y\), and \(n = 20\), then employ the formula to determine each component of our expansion.

Combinatorics

Combinatorics is the area of mathematics that deals with counting, arranging, and analyzing discrete structures. It plays a significant role in calculating the binomial coefficients, which are crucial in the Binomial Expansion. These coefficients show up in the Binomial Theorem as \(\binom{n}{k}\). For an expression \((a + b)^n\), they determine the weight each term contributes to the polynomial. To calculate a binomial coefficient, use the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Where \(!\) denotes the factorial operation, meaning the product of all positive integers up to that number. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). Binomial coefficients are also the entries in Pascal's Triangle, which offers a visual method for determining these coefficients without manual calculation.

Polynomial Expansion

Polynomial Expansion is the process of expressing a binomial like \((x + 2y)^{20}\) as a sum of terms. Each term consists of products of the powers of \(x\) and \(2y\). By employing the Binomial Theorem and the concept of combinatorics, we generate each term's coefficients and respective variable powers. For polynomial expansions, the terms at the beginning often have higher powers of the first variable and lower powers of the second. As you progress through the expansion, the powers of the first term decrease while those of the second increase. This approach of dealing with the powers is important in organizing the expansion, especially when seeking specific terms within the polynomial. In our example, the first three terms of \((x + 2y)^{20}\) were computed as: - First term: \(x^{20}\) - Second term: \(40x^{19}y\) - Third term: \(760x^{18}y^2\) This illustrates how polynomial expansion allows us to accurately compute each term's coefficient and power.