Problem 40 - Algebra and Trigonometry

Question

Find 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: Recall the Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form \((a + b)^n\). It states:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify the expression parameters

In our expression \((\sqrt{2} + y)^{12}\), we have:- \(a = \sqrt{2}\)- \(b = y\)- \(n = 12\)We need to find the term where the power of \(y\) is 3, meaning \(k = 3\).

Step 3: Substitute and simplify the binomial term

The general term in the expansion is given by:\[T_k = \binom{n}{k} a^{n-k} b^k\]Substituting the values, we find the specific term:\[T_3 = \binom{12}{3} (\sqrt{2})^{12-3} y^3\]

Step 4: Calculate the binomial coefficient

Calculate \(\binom{12}{3}\):\[\binom{12}{3} = \frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1} = 220\]

Step 5: Calculate the power of \(\sqrt{2}\)

Compute \((\sqrt{2})^9\):\[(\sqrt{2})^9 = (2)^{9/2} = 2^{4.5} = 2^4 \times 2^{0.5} = 16 \times \sqrt{2}\]

Step 6: Write down the term containing \(y^3\)

Combine all parts of the expression for the term:\[T_3 = 220 \times 16 \times \sqrt{2} \times y^3 = 3520\sqrt{2} y^3\]Thus, the term containing \(y^3\) is \(3520\sqrt{2} y^3\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a crucial element in the Binomial Theorem, which is used to expand expressions involving powers, such as \((a + b)^n\). The binomial coefficient, denoted as \(\binom{n}{k}\), represents the number of ways to choose \k\ items from \ items without regard to order. This coefficient can be calculated using the formula:

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

where \(n!\) or "n factorial" is the product of all positive integers up to \. For example, in the exercise \((\sqrt{2} + y)^{12}\), to find the term with \(y^3\), we calculate \(\binom{12}{3}\). This turns out to be 220, indicating there are 220 different ways to select 3 items from a total of 12.The binomial coefficient helps us derive each term in the binomial expansion, making it a fundamental concept when dealing with polynomials.

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial as a sum of terms. When you expand \((a + b)^n\), you use the Binomial Theorem. The theorem breaks down a power of a sum into a sum of terms involving the products of powers of \a\ and \b\. Each term can be described by:

• \(T_k = \binom{n}{k} a^{n-k} b^k\)

This term represents the contribution of each \(k\)th component of the expansion. In the exercise regarding \(\sqrt{2} + y\), we sought the term with \(y^3\), utilizing the general term formula by substituting in \(k = 3\). Such expansions enable us to write complex expressions as simpler sums, which are easier to manipulate and understand.

Exponentiation

Exponentiation refers to the process of raising a number to a power. In the context of the Binomial Theorem, understanding exponentiation is crucial, especially when dealing with expressions like \((a + b)^n\). This involves raising \(a\) and \(b\) to various powers throughout the expansion. For example, if we take a binomial like \(\sqrt{2} + y\) and seek a particular term such as that containing \(y^3\), it involves computing expressions like \((\sqrt{2})^{12-3}\).Power computations can sometimes involve more complex operations, such as radicals. In this exercise, when calculating \((\sqrt{2})^9\) during the polynomial expansion, it simplifies to \(2^{4.5} = 2^4 \times 2^{0.5} = 16 \times \sqrt{2}\). Understanding how to manipulate exponents helps in successfully applying binomial expansions in mathematics.