Problem 8 - Algebra and Trigonometry

Question

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (1+\sqrt{2})^{6} $$

Step-by-Step Solution

Answer

The expanded expression is \(99 + 70\sqrt{2}\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) are the binomial coefficients. In this exercise, we want to expand \((1+\sqrt{2})^6\). Here, \(a=1\) and \(b=\sqrt{2}\), with \(n=6\).

Step 2: Find Binomial Coefficients

Using Pascal's Triangle, the coefficients for \((a+b)^6\) are: 1, 6, 15, 20, 15, 6, 1. These correspond to the terms of the expansions from \(k=0\) to \(k=6\).

Step 3: Apply Coefficients to Each Term

For each term \(k\), calculate \(\binom{6}{k} a^{6-k} b^k\). This includes substituting \(a=1\) and \(b=\sqrt{2}\) into each term calculated.

Step 4: Calculate Individual Terms

- When \(k=0\): \(\binom{6}{0}(1)^6(\sqrt{2})^0 = 1\).- When \(k=1\): \(\binom{6}{1}(1)^5(\sqrt{2})^1 = 6\sqrt{2}\).- When \(k=2\): \(\binom{6}{2}(1)^4(\sqrt{2})^2 = 15(2) = 30\).- When \(k=3\): \(\binom{6}{3}(1)^3(\sqrt{2})^3 = 20(2\sqrt{2}) = 40\sqrt{2}\).- When \(k=4\): \(\binom{6}{4}(1)^2(\sqrt{2})^4 = 15(4) = 60\).- When \(k=5\): \(\binom{6}{5}(1)^1(\sqrt{2})^5 = 6(4\sqrt{2}) = 24\sqrt{2}\).- When \(k=6\): \(\binom{6}{6}(1)^0(\sqrt{2})^6 = 8\).

Step 5: Combine All Terms

Add all the terms together to find the expanded form: \[1 + 6\sqrt{2} + 30 + 40\sqrt{2} + 60 + 24\sqrt{2} + 8.\]Combine like terms:\[(1 + 30 + 60 + 8) + (6\sqrt{2} + 40\sqrt{2} + 24\sqrt{2}) = 99 + 70\sqrt{2}.\]

Step 6: Final Step: Write the Final Expression

The expanded expression of \((1+\sqrt{2})^6\) is:\[99 + 70\sqrt{2}.\]

Key Concepts

Pascal's Triangle

Pascal's Triangle is an essential tool in understanding and applying the Binomial Theorem. It is a triangular array of numbers where each row represents the coefficients of the expanded form of a binomial expression \((a + b)^n\). To construct Pascal's Triangle:

• Start with a single 1 at the top, known as the apex.
• The numbers on the edges of the triangle are always 1.
• Each internal number in the triangle is formed by adding the two numbers directly above it.

Pascal's Triangle is not just a random set of numbers but a powerful way to visualize binomial coefficients quickly. For instance, in the problem \((1 + \sqrt{2})^6\), the 7th row of Pascal’s Triangle (since rows start from zero) gives the coefficients: 1, 6, 15, 20, 15, 6, and 1.

Binomial Coefficients

Binomial coefficients play a central role in polynomial expansion using the Binomial Theorem. They are denoted as \(\binom{n}{k}\), which represents the number of ways to choose \(k\) items from \(n\) items without regard to order. The formula for calculating a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]However, when using Pascal's Triangle, you can determine these coefficients without needing such calculations, making it faster and more intuitive.
In the exercise of expanding \((1 + \sqrt{2})^6\), we found that the binomial coefficients are 1, 6, 15, 20, 15, 6, 1. These coefficients correspond to terms in the expansion, making it simpler to expand and compute each term accurately.

Polynomial Expansion

Polynomial expansion allows for a compact expression of a binomial raised to a power, to be expanded into a sum of terms. The Binomial Theorem provides a method to achieve this efficiently by using binomial coefficients and powers of the binomial's terms.
When expanding \((1 + \sqrt{2})^6\):

• Start by determining the binomial coefficients using Pascal’s Triangle.
• For each term, multiply the binomial coefficient by the corresponding power of 1 and \(\sqrt{2}\).
• Compute each term separately, such as \(6\sqrt{2}, 30, 40\sqrt{2}\), etc., as demonstrated in the exercise.
• Combine all terms, paying attention to like terms such as those involving \(\sqrt{2}\).

This structured approach simplifies the process, helping prevent errors. Through these steps, the polynomial expansion for \((1 + \sqrt{2})^6\) is derived as \(99 + 70\sqrt{2}\). This provides a clear illustration of how polynomial expansion works.