Problem 21 - Precalculus: Mathematical for Calculus

Question

Use the Binomial Theorcm to expand the expression. $$(x+2 y)^{4}$$

Step-by-Step Solution

Answer

The expanded expression is \(x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = x\), \(b = 2y\), and \(n = 4\). We need to apply this formula to expand the expression.

Step 2: Calculate Binomial Coefficients

The coefficients in the expansion are given by \(\binom{4}{k}\) for \(k = 0, 1, 2, 3, 4\). Calculate these:\[\begin{align*}\binom{4}{0} &= 1, \\binom{4}{1} &= 4, \\binom{4}{2} &= 6, \\binom{4}{3} &= 4, \\binom{4}{4} &= 1.\end{align*}\]

Step 3: Formulate Terms Using the Theorem

For each \(k\) from 0 to 4, use the formula \(\binom{4}{k} x^{4-k} (2y)^k\) to get each term of the expansion:- For \(k=0\): \(\binom{4}{0} x^{4} (2y)^0 = 1 \cdot x^4 \cdot 1 = x^4\)- For \(k=1\): \(\binom{4}{1} x^{3} (2y)^1 = 4 \cdot x^3 \cdot 2y = 8x^3y\)- For \(k=2\): \(\binom{4}{2} x^{2} (2y)^2 = 6 \cdot x^2 \cdot 4y^2 = 24x^2y^2\)- For \(k=3\): \(\binom{4}{3} x^{1} (2y)^3 = 4 \cdot x \cdot 8y^3 = 32xy^3\)- For \(k=4\): \(\binom{4}{4} x^{0} (2y)^4 = 1 \cdot 1 \cdot 16y^4 = 16y^4\)

Step 4: Write the Final Expanded Expression

Combine all the terms to get the final expanded form:\[ \begin{align*}(x + 2y)^4 &= x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4.\end{align*} \]

Key Concepts

Binomial Coefficients

Binomial coefficients represent the numbers that appear in the binomial expansion. They are key in determining the specific weights of each term in the expanded polynomial. For any binomial (\(a + b)^n\) expanded using the Binomial Theorem, the binomial coefficients are calculated using the formula \( \binom{n}{k} \), which is read as "n choose k". This formula is expressed as:

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

This formula calculates how many different ways \(k\) objects can be chosen from \(n\) total objects. In other words, it's all about combinations of terms. In the context of expanding \((x + 2y)^4\), you can see from the solution that the binomial coefficients are:

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

These numbers determine the multipliers for each term as you construct the expanded polynomial.

Polynomial Expansion

Polynomial expansion involves expressing a binomial expression raised to a power as a polynomial sum. When using the Binomial Theorem, it serves as an efficient way to expand expressions like \((x + 2y)^4\). The formula \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) provides a structured way to calculate each term individually. Each term in the expansion is of the form:

• \( \binom{n}{k} a^{n-k} b^k \)

This equation tells us to take products of powers of \(a\) and \(b\), each scaled by the relevant binomial coefficient. In the expansion of \((x + 2y)^4\), the calculation creates five terms, as \(n = 4\) which results in terms from \(k = 0\) to \(k = 4\). Each term is derived by calculating:

• \(1 \cdot x^4 \cdot 1 = x^4\)
• \(4 \cdot x^3 \cdot 2y = 8x^3y\)
• \(6 \cdot x^2 \cdot 4y^2 = 24x^2y^2\)
• \(4 \cdot x \cdot 8y^3 = 32xy^3\)
• \(1 \cdot 1 \cdot 16y^4 = 16y^4\)

These calculated terms are then simply added together to result in the final expanded form.

Combinatorics

Combinatorics is a branch of mathematics that studies the counting, arrangement, and combination of objects in sets. It plays a crucial role when working with the Binomial Theorem since calculating binomial coefficients revolves around combinatorial principles. The essential idea is to determine how many different ways you can form groups of \(k\) elements from a set of \(n\) elements. In our context with \((x + 2y)^4\), understanding that \(\binom{4}{k}\) means "4 choose k" helps conceptualize the task of solving for combinations to calculate these coefficients.

• "4 choose k" represents the number of ways to pick \(k\) elements from 4.
• This application of combinatorics ensures accuracy when determining each term's coefficient in polynomial expansions.

Emphasizing combinatorics makes it easier to understand why each term in an expanded polynomial takes the form it does, as well as recognizing the mathematical resources underpinning the Binomial Theorem's power.