Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Write the binomial expansion for each expression. $$\left(\sqrt{2} r+\frac{1}{m}\right)^{4}$$

Step-by-Step Solution

Answer

The expansion is \(4r^4 + 16\sqrt{2}\frac{r^3}{m} + 12\frac{r^2}{m^2} + 4\sqrt{2}\frac{r}{m^3} + \frac{1}{m^4}\).

Step 1: Identify the Binomial Terms

The expression to be expanded is \((\sqrt{2}r + \frac{1}{m})^4\). Here, \(a = \sqrt{2}r\) and \(b = \frac{1}{m}\). The binomial expression is being raised to the 4th power (\(n = 4\)).

Step 2: Recall Binomial Theorem Formula

The binomial theorem states that \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). We will use this formula to expand \((\sqrt{2}r + \frac{1}{m})^4\).

Step 3: Calculate Each Term of the Expansion

Each term in the expansion is given by \(\binom{4}{k} (\sqrt{2}r)^{4-k} \left(\frac{1}{m}\right)^k\) for \(k = 0, 1, 2, 3,\) and \(4\).- For \(k=0\): \(\binom{4}{0} (\sqrt{2}r)^4 (\frac{1}{m})^0\) = \(1 \cdot (\sqrt{2}r)^4\) = \(4r^4\)- For \(k=1\): \(\binom{4}{1} (\sqrt{2}r)^3 (\frac{1}{m})^1\) = \(4 \cdot \sqrt{2}^3r^3 \cdot \frac{1}{m}\) = \(16\sqrt{2} \frac{r^3}{m}\)- For \(k=2\): \(\binom{4}{2} (\sqrt{2}r)^2 (\frac{1}{m})^2\) = \(6 \cdot 2r^2 \cdot \frac{1}{m^2}\) = \(12 \frac{r^2}{m^2}\)- For \(k=3\): \(\binom{4}{3} (\sqrt{2}r)^1 (\frac{1}{m})^3\) = \(4 \cdot \sqrt{2}r \cdot \frac{1}{m^3}\) = \(4\sqrt{2} \frac{r}{m^3}\)- For \(k=4\): \(\binom{4}{4} (\sqrt{2}r)^0 (\frac{1}{m})^4\) = \(1 \cdot \frac{1}{m^4}\) = \(\frac{1}{m^4}\)

Step 4: Write Out the Complete Expansion

Combine all the calculated terms to write out the full expansion:\[ (\sqrt{2} r + \frac{1}{m})^4 = 4r^4 + 16\sqrt{2}\frac{r^3}{m} + 12\frac{r^2}{m^2} + 4\sqrt{2}\frac{r}{m^3} + \frac{1}{m^4} \]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in mathematics for expanding expressions that are raised to a power, expressed as \((a + b)^n\). This theorem allows us to write the expression in a long form involving sums of terms with powers of 'a' and 'b'. The general formula is:\[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \] Here, \(\binom{n}{k}\) is a binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\). This coefficient determines how many different ways you can choose 'k' items from a total of 'n' items, which is foundational in combinatorial mathematics. The terms \(a^{n-k}\) and \(b^k\) signify the varying powers of 'a' and 'b', that decrease and increase respectively, as 'k' goes from 0 to 'n'. This systematic way of expanding binomials is not only useful in algebra but also in calculus and probability for simplifying polynomial expressions and solving equations.

Combinatorics

Combinatorics is the branch of mathematics concerning the counting, arrangement, and combination of objects. It is heavily used in the binomial expansion through the concept of binomial coefficients. For a given expression \((a + b)^n\), the calculation of binomial coefficients \(\binom{n}{k}\) is crucial. These coefficients tell us how many ways we can pick 'k' items from a collection of 'n' items without considering the sequence.

• \(\binom{n}{0} = 1\) because there's only one way to choose nothing at all.
• \(\binom{n}{k}\) is at its maximum when \(k\) is about \(n/2\), meaning the middle terms in the expansion are often the largest.
• The symmetry in \(\binom{n}{k} = \binom{n}{n-k}\) shows how choosing 'k' items is equivalent to leaving out 'n-k' items.

Combinatorics gives us mathematical tools like permutations, combinations, and the pigeonhole principle, which have far-reaching applications in different fields including computer science, statistics, and optimization problems.

Polynomial Expansion

Polynomial expansion involves expressing a binomial, which is a polynomial with two terms, in an expanded form of multiple terms. Each term in a polynomial is characterized by coefficients and variables raised to certain powers. Using the binomial theorem, we can expand expressions like \((a + b)^n\) into a sum of terms consisting of coefficients (determined by the binomial coefficients) and powers of the initial two terms. In our original exercise, the expansion of \((\sqrt{2} r + \frac{1}{m})^4\) into a polynomial results in separate terms like \(4r^4\), \(16\sqrt{2}\frac{r^3}{m}\), and so forth.
This decomposition not only allows us to simplify complex expressions into manageable components but also helps in performing operations like differentiation or integration in calculus. Understanding polynomial expansion gives insight into how functions behave and helps in approximating values, solving equations, and analyzing mathematical models.