Problem 40 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. The term containing \(y^{3}\) in the expansion of \((\sqrt{2}+y)^{12}\).

Step-by-Step Solution

Answer

The term containing \( y^3 \) is \( 3520 \sqrt{2} y^3 \).

Step 1: Determine the General Term Formula

In the binomial expansion of \((a + b)^n\), the general term (\(T_k\)) is given by: \[ T_k = \binom{n}{k} a^{n-k} b^k \] where \( n \) is the power of the binomial, \( a \) is the first term, \( b \) is the second term, and \( k \) is the term number starting at 0.

Step 2: Identify Variables from Problem

In this problem, \( a = \sqrt{2} \), \( b = y \), and \( n = 12 \). We want the term that contains \( y^3 \), therefore \( k = 3 \).

Step 3: Apply Values to General Term Formula

Apply these values to the general term formula: \[ T_3 = \binom{12}{3} (\sqrt{2})^{12-3} y^3 \] which simplifies to: \[ T_3 = \binom{12}{3} (\sqrt{2})^9 y^3 \]

Step 4: Calculate the Binomial Coefficient

The binomial coefficient is calculated as: \[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

Step 5: Simplify the Term

Calculate \((\sqrt{2})^9\): \[ (\sqrt{2})^9 = (2^{1/2})^9 = 2^{9/2} = 2^4 \times \sqrt{2} = 16\sqrt{2} \]. So, the term becomes: \[ T_3 = 220 \times 16\sqrt{2} \times y^3 \].

Step 6: Final Calculation

Multiply and simplify: \[ T_3 = 220 \times 16\sqrt{2} \times y^3 = 3520 \sqrt{2} y^3 \].

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that describes the expansion of a binomial expression raised to any non-negative integer power. If you have an expression of the form \((a + b)^n\), the theorem tells you how to expand it into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\). Here are some key points about the theorem:

• The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements, signifying the weight of each term in the expansion.
• The variables \(a\) and \(b\) are the terms of the binomial being expanded, and \(n\) is the power to which the binomial is raised.
• The expansion results in \(n+1\) terms, beginning from \(k=0\) to \(k=n\).

By applying the Binomial Theorem, it's actually possible to calculate any specific term in the expansion without having to expand the entire expression. This is particularly useful for large powers where full expansion would be cumbersome.

General Term Formula

In the realm of the binomial expansion, the General Term Formula is a vital tool for pinpointing any specific term without unravelling the entire expansion. It is mathematically described as:\[T_k = \binom{n}{k} a^{n-k} b^k\]In this formula:

• \(T_k\) stands for the \(k\)-th term in the expansion.
• \(\binom{n}{k}\) is the binomial coefficient.
• \(a\) and \(b\) are the terms from the binomial expression \((a + b)^n\).
• \(n\) is the total power of the binomial expansion.
• \(k\) signifies the specific term number starting at zero.

This formula is essential when you're tasked with finding terms, like those involving a particular variable to a specific power. For instance, if you're asked to find a term involving \(y^3\) in an equation, you'd set \(k\) so that \(b^k = y^3\) and solve for the remaining variables accordingly.

Exponents

Exponents are another central concept when dealing with binomial expansions. They represent how many times a number, known as the base, is multiplied by itself. For example, in the expression \((\sqrt{2})^9\), \(\sqrt{2}\) is the base, and \(9\) is the exponent. Here are some important points about exponents:

• Exponentiation is a shorthand way of denoting repeated multiplication.
• In binomial expansions, exponents apply to the terms \(a^{n-k}\) and \(b^k\).
• Exponents can be whole numbers, fractions, or even negative numbers;
• Fractional exponents, such as \(2^{9/2}\), denote roots and powers, where \(2^{9/2} = (2^4) \times \sqrt{2}\).

Understanding how to manipulate exponents is critical for simplifying terms in binomial expansions, as shown when computing \((\sqrt{2})^9\) to get \(16\sqrt{2}\). Mastery of this topic allows you to tackle various problems involving powers and roots efficiently.