Problem 12 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determine the Row of Pascal's Triangle

To expand \( (1+\sqrt{2})^{6} \), we need the 6th row of Pascal's Triangle. Recall that the rows start at 0, so the 6th row is actually the 7th row of natural numbers.The 6th row is: \( 1, 6, 15, 20, 15, 6, 1 \).

Step 2: Write the General Term of the Binomial Expansion

The binomial expansion formula is:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]For \( (1+\sqrt{2})^{6} \), set \( a = 1 \) and \( b = \sqrt{2} \) with \( n = 6 \).

Step 3: Apply the Binomial Coefficients

Using the 6th row coefficients from Pascal's Triangle: \( 1, 6, 15, 20, 15, 6, 1 \), we calculate each term of the expansion:- For \( k = 0 \): \( 1 \cdot (1)^6 \cdot (\sqrt{2})^0 \)- For \( k = 1 \): \( 6 \cdot (1)^5 \cdot (\sqrt{2})^1 \)- For \( k = 2 \): \( 15 \cdot (1)^4 \cdot (\sqrt{2})^2 \)- For \( k = 3 \): \( 20 \cdot (1)^3 \cdot (\sqrt{2})^3 \)- For \( k = 4 \): \( 15 \cdot (1)^2 \cdot (\sqrt{2})^4 \)- For \( k = 5 \): \( 6 \cdot (1)^1 \cdot (\sqrt{2})^5 \)- For \( k = 6 \): \( 1 \cdot (1)^0 \cdot (\sqrt{2})^6 \)

Step 4: Simplify Each Term

Calculate each term:- \( (1)^6 \cdot (\sqrt{2})^0 = 1 \)- \( 6 \cdot (\sqrt{2}) = 6\sqrt{2} \)- \( 15 \cdot 2 = 30 \)- \( 20 \cdot 2\sqrt{2} = 40\sqrt{2} \)- \( 15 \cdot 4 = 60 \)- \( 6 \cdot 4\sqrt{2} = 24\sqrt{2} \)- \( 1 \cdot 8 = 8 \)

Step 5: Combine Terms to Get the Expanded Form

Add the simplified terms together:\( 1 + 6\sqrt{2} + 30 + 40\sqrt{2} + 60 + 24\sqrt{2} + 8 \).Combine like terms (integers and terms with \( \sqrt{2} \)):- Integers: \( 1 + 30 + 60 + 8 = 99 \)- Terms with \( \sqrt{2} \): \( 6\sqrt{2} + 40\sqrt{2} + 24\sqrt{2} = 70\sqrt{2} \).Final expanded form is:\( 99 + 70\sqrt{2} \).

Key Concepts

Binomial Expansion

Binomial expansion is a powerful algebraic method used to expand expressions of the form \((a + b)^n\). This technique allows us to express the power of a binomial as a sum of terms involving powers of both variables, along with coefficients that depend on the order of expansion. The central idea is to apply the Binomial Theorem, which provides a formula:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

The formula expresses the binomial expansion as a series of terms, where each term is a product of a binomial coefficient, a power of \(a\), and a power of \(b\).
This method is particularly useful because it simplifies the work needed to expand large powers of binomials. The power \(n\) indicates the number of terms in the expansion. For example, when expanding \((1+\sqrt{2})^6\), we find that there are 7 terms in total since the binomial expansion will include powers from 0 to 6.
Each term in the expansion can be calculated by finding the appropriate binomial coefficient and then multiplying it by the respective powers of \(a\) and \(b\). This makes binomial expansion particularly efficient compared to straightforward multiplication, especially for large values of \(n\).

Binomial Coefficients

Binomial coefficients are a key component of binomial expansion. They tell us how to weight the different terms in the expansion and are represented by \(\binom{n}{k}\), which is read as "n choose k". Each binomial coefficient corresponds to the number of ways to choose \(k\) elements out of \(n\) total elements, regardless of order.
An important property of binomial coefficients is that they form what is known as Pascal's Triangle. Each row of Pascal's Triangle corresponds to a set of binomial coefficients for a particular power \(n\). For instance, to expand \((1 + \sqrt{2})^6\), we used the coefficients: 1, 6, 15, 20, 15, 6, and 1.

• The \(k\)-th entry in the \(n\)-th row is given by \(\binom{n}{k}\).
• The sum of the elements in the \(n\)-th row also represents \(2^n\).

When you use Pascal's Triangle for binomial expansion, it significantly simplifies the process of finding each coefficient, making it easier to write down the expansion quickly and accurately. The symmetrical nature of the triangle also means some coefficients are repeated, which helps in verifying results intuitively.

Precalculus

Precalculus is a foundation course that prepares students for calculus. It brings together various mathematical concepts like algebra, geometry, and analysis. Binomial expansion is a crucial topic in precalculus, as it combines algebraic techniques with combinatorial principles.
Understanding binomial coefficients and expansions is vital before students embark on calculus, as these topics frequently appear in limits, series, and polynomial approximations. In precalculus, students learn to apply the Binomial Theorem as it sets the stage for more complex mathematical rules and theorems in calculus.

• Key concepts like functions, equations, and graphing build a robust mathematical framework.
• Mastery of these ideas ensures students can handle calculus's abstract concepts and differential and integral calculus.

Binomial expansion showcases how algebra and combinatorics intertwine to solve complex expressions efficiently, preparing students for the logical thinking required in advanced math courses.