Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Write the binomial expansion for each expression. $$(p+2 q)^{4}$$

Step-by-Step Solution

Answer

The binomial expansion is \(p^4 + 8p^3q + 24p^2q^2 + 32pq^3 + 16q^4\).

Step 1: Identify Binomial Components

In the expression \((p + 2q)^{4}\), identify \(p\) as the first term and \(2q\) as the second term. The exponent is 4.

Step 2: Recall Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Here, \(a = p\), \(b = 2q\), and \(n = 4\).

Step 3: Compute Each Term

Calculate term by term using the binomial theorem. This involves calculating \(\binom{4}{k} p^{4-k} (2q)^k\) for each \(k\) from 0 to 4. Here are the steps:1. For \(k = 0\): \(\binom{4}{0} p^{4} (2q)^0 = p^4\)2. For \(k = 1\): \(\binom{4}{1} p^{3} (2q)^1 = 4p^3 \times 2q = 8p^3q\)3. For \(k = 2\): \(\binom{4}{2} p^{2} (2q)^2 = 6p^2 \times 4q^2 = 24p^2q^2\)4. For \(k = 3\): \(\binom{4}{3} p^{1} (2q)^3 = 4p \times 8q^3 = 32pq^3\)5. For \(k = 4\): \(\binom{4}{4} p^{0} (2q)^4 = 1 \times 16q^4 = 16q^4\).

Step 4: Write the Final Expansion

Combine all the terms calculated in Step 3:- The expansion is: \(p^4 + 8p^3q + 24p^2q^2 + 32pq^3 + 16q^4\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful algebraic tool that allows us to expand expressions raised to a power in a systematic way. It's particularly useful for expressions like \((a + b)^n\). Discovered by Isaac Newton, this theorem gives us a formula:

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

This formula means that we can find each term in the expansion by looping through all values from \(k = 0\) to \(k = n\). Each term is the product of a binomial coefficient, a power of \(a\), and a power of \(b\).
For the problem \((p + 2q)^4\), \(a = p\), \(b = 2q\), and \(n = 4\). Therefore, the expansion involves computing terms from the sum specified by the binomial theorem.

Binomial Coefficients

Understanding binomial coefficients is key to utilizing the binomial theorem. Binomial coefficients, represented as \(\binom{n}{k}\), tell us how many ways we can choose \(k\) items from \(n\) items, where the order of selection does not matter.
The formula to calculate this is:

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

where \(n!\) is the factorial of \(n\), which is the product of all positive integers up to \(n\).
For example, in our exercise, with \((p + 2q)^4\), to find the \(k^{th}\) term, we use coefficients like \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3},\) and \(\binom{4}{4}\). These coefficients tell us the number of ways terms can be combined and are an essential part of the polynomial expansion.

Polynomial Expansion

In mathematics, polynomial expansion is the process of expressing a power of a binomial expression as a sum of terms with coefficients. Each term is a product of the binomial coefficients and powers of the original binomial components.
In the binomial expression \((p + 2q)^4\), expansion means converting it into a polynomial, which here results in:

• \(p^4\)
• \(+ 8p^3q\)
• \(+ 24p^2q^2\)
• \(+ 32pq^3\)
• \(+ 16q^4\)

Each term in this expansion not only contains the variables \(p\) and \(q\) in different powers but also includes the binomial coefficient and a power of 2 from the second term \((2q)\).
This process is immensely useful for simplifying calculations and expressing functions in a more manageable form, especially in statistics and probability.