Problem 25 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$(p-q)^{5}$$

Step-by-Step Solution

Answer

\((p-q)^5 = p^5 - 5p^4q + 10p^3q^2 - 10p^2q^3 + 5pq^4 - q^5\).

Step 1: Understand the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form \((a + b)^n\). The expansion is given by \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify the Components of the Binomial Expression

In the expression \((p - q)^5\), identify the components: \(a = p\), \(b = -q\), and \(n = 5\).

Step 3: Apply the Binomial Theorem to the Expression

Apply the binomial theorem to expand \((p - q)^5\). Substitute \(a = p\), \(b = -q\), and \(n = 5\) into the binomial expansion formula. The expansion will have terms from \(k=0\) to \(k=5\).

Step 4: Calculate Each Term in the Expansion

For each value of \(k\) from 0 to 5, calculate:- The binomial coefficient \(\binom{5}{k}\).- The power of \(a\) which is \(p^{5-k}\).- The power of \(b\) which is \((-q)^k\).- Multiply these components together for each term in the expansion.

Step 5: Write out the Full Expansion

Combine all calculated terms to write out the full expansion:\((p - q)^5 = \binom{5}{0}p^5(-q)^0 + \binom{5}{1}p^4(-q)^1 + \binom{5}{2}p^3(-q)^2 + \binom{5}{3}p^2(-q)^3 + \binom{5}{4}p^1(-q)^4 + \binom{5}{5}p^0(-q)^5\).

Step 6: Simplify Each Term

Simplify the terms:1. \(\binom{5}{0}p^5 = 1 \cdot p^5 = p^5\)2. \(\binom{5}{1}p^4(-q) = 5p^4(-q) = -5p^4q\)3. \(\binom{5}{2}p^3(q^2) = 10p^3q^2\)4. \(\binom{5}{3}p^2(-q^3) = -10p^2q^3\)5. \(\binom{5}{4}p(-q^4) = 5pq^4\)6. \(\binom{5}{5}(-q)^5 = -q^5\)\Combine the terms: \(p^5 - 5p^4q + 10p^3q^2 - 10p^2q^3 + 5pq^4 - q^5\).

Key Concepts

Binomial Theorem

The binomial theorem is an essential tool in algebra for expanding expressions raised to a power. It provides a powerful method to systematically expand expressions like \((a + b)^n\) without manually multiplying the expressions repeatedly. This theorem is represented as a formula:

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

This generates each term in the expansion by using specific coefficients, powers, and combinations. The binomial theorem helps simplify the work involved in polynomial expansion, enabling students to understand higher powers of binomials efficiently.

Binomial Coefficient

At the heart of the binomial theorem and polynomial expansions are the binomial coefficients, denoted as \(\binom{n}{k}\). These coefficients are calculated based on combinations, indicating how many ways we can choose \(k\) elements from a set of \(n\) elements. The formula for a binomial coefficient is:

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

In the expansion of \((p - q)^5\), you will see these coefficients as essential parts forming each term, determining how the powers of \(p\) and \(q\) interplay. They ensure the expansion's terms are consistent with combinatorial principles.

Polynomial Expansion

Polynomial expansion involves turning a compact, exponential expression like \((x - y)^n\) into a longer polynomial form exhibiting individual terms. Using the binomial theorem, the polynomial expansion unfolds systematically. Each term in the expansion combines:

• The binomial coefficient \(\binom{n}{k}\).
• The power of the first term, e.g., \(a^{n-k}\).
• The power of the second term, e.g., \(b^k\).

This process simplifies calculations and provides insights into the relationships and contributions of each part of the original binomial expression to its expanded form.

Algebraic Expression

Binomial expressions such as \((a - b)^n\) are specific types of algebraic expressions involving two terms. Algebraic expressions cover a broad range of forms, from simple monomials to complex polynomials. They represent quantities in mathematical terms, allowing operations such as addition, subtraction, multiplication, and division among algebraic terms.In the case of binomial expressions, expansion allows for transforming the expression into an equivalent polynomial form, providing a clearer view of each component's contribution to the value and behavior of the expression across varying input values.