Problem 71 - Algebra 2

Question

Expand each binomial. $$ (p+q)^{6} $$

Step-by-Step Solution

Answer

The expansion of \((p+q)^{6}\) using the binomial theorem will result in \(p^{6} + 6p^{5}q + 15p^{4}q^{2} + 20p^{3}q^{3} + 15p^{2}q^{4} + 6pq^{5} + q^{6}\)

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer \(n\), the expansion of the \(n^{th}\) power of a binomial \( (p+q)^{n} \) is given by \( (p + q)^n = \sum_{k=0}^{n} {n \choose k} p^{n-k} q^{k} \). In this case, \(n=6\).

Step 2: Calculate the binomial coefficients

Using Python, we can calculate the binomial coefficients, which are {6\choose 0}, {6\choose 1}, {6\choose 2}, {6\choose 3}, {6\choose 4}, {6\choose 5}, {6\choose 6} which are equal to 1, 6, 15, 20, 15, 6, 1 respectively.

Step 3: Apply the Binomial Theorem

Substitute these numbers back into the binomial theorem along with the \(p\) and \(q\) terms to expand the expression. The expanded polynomial is \(p^{6} + 6p^{5}q + 15p^{4}q^{2} + 20p^{3}q^{3} + 15p^{2}q^{4} + 6pq^{5} + q^{6}\).

Key Concepts

Binomial Expansion

The concept of binomial expansion is a key component in algebra and calculus, primarily used to expand expressions that are raised to a power. When you hear "binomial," think of expressions of the form \((p + q)^n\), where a binomial consists of two terms. The binomial expansion involves expressing this as a sum of multiple terms. Each term in the expansion has specific coefficients, exponents, and demonstrates the number of possible ways an event can occur.

• In essence, binomial expansion is about transforming a compact binomial into a lengthy expression of terms.
• Each term has a coefficient, which can be calculated using the binomial theorem.

Binomial expansion simplifies calculations by breaking down expressions into manageable bits. When expanded, as shown in the original exercise with \((p+q)^6\), it clearly shows how higher powers of expressions distribute themselves across multiple terms.

Binomial Coefficients

Every term in a binomial expansion has a unique binomial coefficient. These coefficients play an essential role by indicating the number of ways to choose elements from a set, sometimes called 'combinations'.

• The formula \( {n \choose k} \) is used to find these coefficients. It represents the number of ways to select \(k\) items from \(n\) items without considering the order.
• For instance, in our problem, the coefficients for \((p+q)^6\) are \(1, 6, 15, 20, 15, 6, 1\).

Finding these coefficients helps decode the potential multiplicative factors each element in the expanded expression terms will have. Remember that these coefficients are symmetrical, providing balance and pattern recognizable in Pascal’s Triangle.

Polynomial Expansion

When binomials like \((p+q)^6\) are expanded, they turn into a polynomial with multiple terms. Each term includes a product of the variables raised to different powers and is determined using the binomial theorem.

• Polynomial expansions turn concise expressions into a series with broader terms that illustrate functions, statistical distributions, and more.
• The original exercise provides an example of a polynomial expansion: \(p^{6} + 6p^{5}q + 15p^{4}q^{2} + 20p^{3}q^{3} + 15p^{2}q^{4} + 6pq^{5} + q^{6}\).

Notice how in each term, the sum of the exponents of \(p\) and \(q\) equals the original exponent in the binomial (6, in this case). This transformation is crucial when operating with polynomials, simplifying complex expressions.

Mathematical Theorems

Mathematical theorems like the binomial theorem provide fundamental rules that apply universally to specific calculations and operations. The binomial theorem, in particular, lays down guidelines for expanding expressions raised to a power.

• It provides a formula that is applicable to any positive integer: \((p + q)^n = \sum_{k=0}^{n} {n \choose k} p^{n-k} q^{k}\).
• This theorem is essential in various fields including, but not limited to, probability, statistics, and computational mathematics.

Understanding mathematical theorems helps us predict outcomes, understand structure, and derive new insights from existing knowledge. They create shortcuts for complex operations, allowing mathematicians and scientists to build on proven concepts easily.