Problem 39 - Precalculus Mathematics for Calculus

Question

Find the term containing \(x^{4}\) in the expansion of \((x+2 y)^{10}\).

Step-by-Step Solution

Answer

The term containing \(x^{4}\) is \(13440 x^4 y^6\).

Step 1: Identify the Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, we have \((x + 2y)^{10}\), where \(a = x\), \(b = 2y\), and \(n = 10\).

Step 2: Determine General Term Structure

Using the binomial theorem, the general term \(T_k\) in the expansion of \((x + 2y)^{10}\) can be expressed as \(T_k = \binom{10}{k} x^{10-k} (2y)^k\).

Step 3: Set up for the Desired Power of x

We are seeking the term where the power of \(x\) is 4. Therefore, we set up the equation: \(10 - k = 4\). This simplifies to \(k = 6\).

Step 4: Substitute k into the General Term

Substitute \(k = 6\) into the general term: \(T_6 = \binom{10}{6} x^{10-6} (2y)^6\). Simplify this to \(T_6 = \binom{10}{6} x^{4} (2^6) y^6\).

Step 5: Calculate Binomial Coefficient and Power of 2

Calculate \(\binom{10}{6} = 210\) and \(2^6 = 64\).

Step 6: Simplify the Term

Substitute the values from Step 5 into the expression for \(T_6\): \[ T_6 = 210 x^4 (64) y^6 = 13440 x^4 y^6.\] Thus, the term containing \(x^4\) is \(13440 x^4 y^6\).

Key Concepts

Polynomial Expansion

When dealing with polynomial expansion, particularly through the Binomial Theorem, we are stretching expressions of the form \((a + b)^n\) into a sum of multiple terms. This field of algebra simplifies the complexity of raising binomials to powers by providing a structured approach. In our exercise, we are expanding \((x + 2y)^{10}\). Through expansion, each term showcases a specific combination of the components \(a\) and \(b\), alongside coefficients. These terms show varying powers, which add up to \(n\), the original exponent.
The Binomial Theorem is central for polynomial expansion. It helps to predict each term's structure without manually calculating each power. This saves time and avoids errors, especially when \(n\) is large. Engage with it by identifying the variables needed and applying them to the theorem's formula form.

Binomial Coefficients

Binomial coefficients are numerical factors that arise in the expansion using the Binomial Theorem. Displayed as \(\binom{n}{k}\), these coefficients denote the number of ways to choose \(k\) items from \(n\) items without concern for the order selected. For example, \(\binom{10}{6}\) is part of our calculation, representing the sixth term's coefficient in the expansion.
To compute binomial coefficients, the formula is simple:

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

Here, \(!\) represents factorial, meaning the product of an integer and all the integers below it. These coefficients not only determine the weight each element in a term contributes but are also linked to combinatorial ideas in mathematics.For our exercise, understanding binomial coefficients is essential as they determine the multiplier for the combined terms.

Mathematics Problem Solving

Mathematics problem solving involves tackling a problem methodically and strategically. This requires grasping the core concepts involved, identifying patterns, and applying theories such as the Binomial Theorem. In our exercise, we faced finding specifically the term containing \(x^{4}\) in the polynomial expansion of \((x + 2y)^{10}\).
To solve such a task:

• Understand the structure offered by the binomial theorem.
• Determine the general formula for any term in the expansion.
• Concentrate on the conditions you need, such as the required power of \(x\), translating them into equations.
• Solve these equations to pinpoint exact term information.

Carefully following these steps ensures accurate and efficient solutions. Mathematical problem-solving, whether in polynomial expansions or other areas, often requires a blend of theoretical knowledge and practical application. A systematic approach removes the guesswork, leading to clear and valid answers.