Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Use the binomial theorem to expand and simplify. $$ 5\left(\frac{1}{2} x+y^{3}\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form is \( \frac{5}{16}x^4 + \frac{5}{2}x^3y^3 + \frac{15}{2}x^2y^6 + 10xy^9 + 5y^{12} \).

Step 1: Identify the Variables

In the expression \( 5\left(\frac{1}{2} x + y^{3}\right)^{4} \), identify \( a = \frac{1}{2}x \) and \( b = y^3 \). We will expand \( (a + b)^4 \).

Step 2: Apply the Binomial Theorem

The binomial theorem states: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \]For \( n = 4 \), substitute the values:\[ (\frac{1}{2} x + y^3)^4 = \sum_{k=0}^{4} \binom{4}{k} (\frac{1}{2}x)^{4-k} (y^3)^{k} \].

Step 3: Calculate Each Term in the Expansion

Compute each term separately:- When \( k=0 \): \[ \binom{4}{0} (\frac{1}{2}x)^4 (y^3)^{0} = 1 \cdot (\frac{1}{2}x)^4 = \frac{1}{16}x^4 \]- When \( k=1 \): \[ \binom{4}{1} (\frac{1}{2}x)^3 (y^3)^1 = 4 \cdot \frac{1}{8}x^3 \cdot y^3 = \frac{1}{2}x^3y^3 \]- When \( k=2 \): \[ \binom{4}{2} (\frac{1}{2}x)^2 (y^3)^2 = 6 \cdot \frac{1}{4}x^2 \cdot y^6 = \frac{3}{2}x^2y^6 \]- When \( k=3 \): \[ \binom{4}{3} (\frac{1}{2}x)^1 (y^3)^3 = 4 \cdot \frac{1}{2}x \cdot y^9 = 2xy^9 \]- When \( k=4 \): \[ \binom{4}{4} (\frac{1}{2}x)^0 (y^3)^4 = 1 \cdot y^{12} = y^{12} \].

Step 4: Combine the Terms

Add these terms together to get the expanded form:\[ \frac{1}{16}x^4 + \frac{1}{2}x^3y^3 + \frac{3}{2}x^2y^6 + 2xy^9 + y^{12} \].

Step 5: Multiply by the Constant

Now multiply the expanded form by 5:\[ 5 \left( \frac{1}{16}x^4 + \frac{1}{2}x^3y^3 + \frac{3}{2}x^2y^6 + 2xy^9 + y^{12} \right) = \frac{5}{16}x^4 + \frac{5}{2}x^3y^3 + \frac{15}{2}x^2y^6 + 10xy^9 + 5y^{12} \].

Key Concepts

Polynomial Expansion

Polynomial expansion involves breaking down expressions into simplified forms. The Binomial Theorem is a powerful tool used in polynomial expansion. It allows us to expand expressions of the form \( (a + b)^n \) into a sum of terms involving powers of \( a \) and \( b \). For our equation \( 5\left(\frac{1}{2}x + y^3\right)^4 \), the goal is to expand the polynomial by determining each term in its expanded form.
This involves identifying the base components, \( a = \frac{1}{2}x \) and \( b = y^3 \), and using the theorem to calculate the expanded expression. The binomial expansion expresses \( (a + b)^4 \) as a sum of multiple terms each raised to a power and multiplied by a coefficient. Each term in the expansion comes from multiplying out the expression \( (a + b) \) four times, a process facilitated by the Binomial Coefficients.

Binomial Coefficients

Binomial coefficients are a crucial aspect of applying the Binomial Theorem. They provide the values necessary to determine how much each term of the expanded polynomial is scaled. The coefficients are represented as \( \binom{n}{k} \) and are calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
This formula calculates the number of ways to choose \( k \) items from \( n \) items without regard to order. In the context of polynomial expansion, \( n \) is the power of the binomial and \( k \) changes from 0 to \( n \).
In our case, for expanding \( (\frac{1}{2}x + y^3)^4 \), the coefficients \( \binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4} \) must be calculated. These coefficients help scale each term, determining how the components \( (\frac{1}{2} x)^{4-k} \) and \( (y^3)^{k} \) contribute to the overall expression.

Algebraic Simplification

Algebraic simplification is the process of finding a simpler form for an expression. After expanding the binomial \( \left(\frac{1}{2}x + y^3\right)^4 \) using binomial coefficients, we simplify individual terms and combine them. Each term is simplified by raising the powers and multiplying coefficients.

For example, a term such as \( \binom{4}{1} \left(\frac{1}{2}x\right)^3 (y^3)^1 \) becomes \( 4 \times \frac{1}{8}x^3 \times y^3 \) which simplifies to \( \frac{1}{2}x^3y^3 \). This simplification ensures that the final result is as simple and clear as possible.

• Handling powers of \( x \) and \( y \)
• Multiplying coefficients together properly

Finally, the entire expression is multiplied by 5, as per the original equation. The result: \( \frac{5}{16}x^4 + \frac{5}{2}x^3y^3 + \frac{15}{2}x^2y^6 + 10xy^9 + 5y^{12} \). This step ensures a complete and simple solution to the polynomial expansion.