Problem 31 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$(7 p+2 q)^{4}$$

Step-by-Step Solution

Answer

The expanded form is \(2401p^4 + 2744p^3q + 1176p^2q^2 + 224pq^3 + 16q^4\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer \(n\), \((a + b)^n\) is expanded as \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]where \(\binom{n}{k}\) is the binomial coefficient equal to \(\dfrac{n!}{k!(n-k)!}\).

Step 2: Identify Components for Expansion

For the expression \((7p + 2q)^4\), identify \(a = 7p\) and \(b = 2q\), and \(n = 4\).

Step 3: Calculate Binomial Coefficients

Calculate binomial coefficients \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \text{and} \ \binom{4}{4}\). These values are:- \(\binom{4}{0} = 1\)- \(\binom{4}{1} = 4\)- \(\binom{4}{2} = 6\)- \(\binom{4}{3} = 4\)- \(\binom{4}{4} = 1\).

Step 4: Apply Binomial Expansion Formula

Substitute \(a = 7p\), \(b = 2q\), and calculated binomial coefficients into the binomial expansion formula:\[(7p + 2q)^4 = \binom{4}{0}(7p)^4(2q)^0 + \binom{4}{1}(7p)^3(2q)^1 + \binom{4}{2}(7p)^2(2q)^2 + \binom{4}{3}(7p)^1(2q)^3 + \binom{4}{4}(7p)^0(2q)^4\]

Step 5: Simplify Each Term

Calculate and simplify each term from the expansion:- First term: \(\binom{4}{0}(7p)^4(2q)^0 = 1 \cdot 2401p^4 \cdot 1 = 2401p^4\)- Second term: \(\binom{4}{1}(7p)^3(2q)^1 = 4 \cdot 343p^3 \cdot 2q = 2744p^3q\)- Third term: \(\binom{4}{2}(7p)^2(2q)^2 = 6 \cdot 49p^2 \cdot 4q^2 = 1176p^2q^2\)- Fourth term: \(\binom{4}{3}(7p)^1(2q)^3 = 4 \cdot 7p \cdot 8q^3 = 224pq^3\)- Fifth term: \(\binom{4}{4}(7p)^0(2q)^4 = 1 \cdot 1 \cdot 16q^4 = 16q^4\).

Step 6: Write the Final Expanded Expression

Combine all simplified terms to get the expanded form: \[(7p + 2q)^4 = 2401p^4 + 2744p^3q + 1176p^2q^2 + 224pq^3 + 16q^4\].

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool used to expand expressions of the form \((a + b)^n\). This theorem tells us that each term in the expansion is given by the formula:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]Here, \(\binom{n}{k}\) represents the binomial coefficient, which determines the weight of each term in the expansion. It is crucial in algebra for solving polynomial expressions quickly.

To effectively use the Binomial Theorem:

• Identify \(a\), \(b\), and \(n\) in your expression.
• Calculate the binomial coefficients for each term.
• Substitute into the formula and simplify each term.

For example, in the expression \((7p + 2q)^4\), \(a = 7p\), \(b = 2q\), and \(n = 4\). The expansion will involve calculating and putting together each term using these values.

Polynomial Expansion

Polynomial Expansion involves expressing a polynomial raised to an exponent as a sum of simpler terms. The Binomial Theorem aids in finding these component terms without expanding manually.

When expanding a polynomial like \((7p + 2q)^4\), each term is derived from:

• The binomial coefficient \(\binom{n}{k}\), which dictates the weight of each term
• Powers of \(a\) and \(b\) adjusted by their positions \(k\) and \(n-k\), respectively.

This systematic approach allows us to break down the complexity of higher powers step by step. For instance, the terms of the expression \((7p + 2q)^4\) are calculated as

• \(2401p^4\),
• \(2744p^3q\),
• \(1176p^2q^2\),
• \(224pq^3\),
• \(16q^4\)

resulting in a simplified expansion.

Binomial Coefficients

Binomial Coefficients are key in determining the distribution of terms in a binomial expansion. Denoted as \(\binom{n}{k}\) and often called "n choose k," these coefficients are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(n!\) (n factorial) is the product of all positive integers up to \(n\).

In a binomial expansion, each coefficient represents the unique number of ways to select \(k\) items from \(n\) options, and this directly affects the size of each term. For the expansion of \((7p + 2q)^4\), the required coefficients are:

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

Understanding and calculating these coefficients are essential for constructing the full polynomial expansion effectively.