Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Write the binomial expansion for each expression. $$\left(\frac{1}{k}-\sqrt{3} p\right)^{3}$$

Step-by-Step Solution

Answer

The expansion is \( \frac{1}{k^3} - \frac{3\sqrt{3}p}{k^2} + \frac{9p^2}{k} - 3\sqrt{3}p^3 \).

Step 1: Identify the Expression Components

The given expression is \( \left( \frac{1}{k} - \sqrt{3}p \right)^3 \). Here, \( a = \frac{1}{k} \) and \( b = -\sqrt{3}p \).

Step 2: Recall Binomial Theorem

The binomial theorem states that \( (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \). We will use this to expand the given expression.

Step 3: Apply Binomial Theorem

For \( n = 3 \), the expansion is:\[\sum_{r=0}^{3} \binom{3}{r} \left(\frac{1}{k}\right)^{3-r} (-\sqrt{3}p)^r \].

Step 4: Calculate Coefficients for Each Term

Calculate each term of the binomial expansion separately:- For \( r = 0 \): \[ \binom{3}{0} \left(\frac{1}{k}\right)^3 (-\sqrt{3}p)^0 = \left(\frac{1}{k}\right)^3 = \frac{1}{k^3} \]- For \( r = 1 \): \[ \binom{3}{1} \left(\frac{1}{k}\right)^{2} (-\sqrt{3}p)^1 = -3 \times \frac{1}{k^2} \times \sqrt{3}p = -\frac{3\sqrt{3}p}{k^2} \]- For \( r = 2 \): \[ \binom{3}{2} \left(\frac{1}{k}\right)^{1} (-\sqrt{3}p)^2 = 3 \times \frac{1}{k} \times 3p^2 = \frac{9p^2}{k} \]- For \( r = 3 \): \[ \binom{3}{3} \left(\frac{1}{k}\right)^0 (-\sqrt{3}p)^3 = - (\sqrt{3}p)^3 = -3\sqrt{3}p^3 \]

Step 5: Construct the Binomial Expansion

Combine all of the terms to construct the binomial expansion:\[\left(\frac{1}{k} - \sqrt{3}p \right)^3 = \frac{1}{k^3} - \frac{3\sqrt{3}p}{k^2} + \frac{9p^2}{k} - 3\sqrt{3}p^3\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool that allows us to expand expressions that are raised to a power, specifically binomials. A binomial is any expression with two terms, like \( (a + b)^n \). The theorem expresses this expansion as a sum of terms involving binomial coefficients. These coefficients are seen in Pascal's Triangle and are calculated using combinations, represented as \( \binom{n}{r} \). Each term in the expansion takes the form of \( \binom{n}{r} a^{n-r} b^r \), where \( a \) and \( b \) are the terms of the binomial and \( r \) runs from zero to \( n \).
To apply this to our specific expression \( \left( \frac{1}{k} - \sqrt{3}p \right)^3 \), we identify it fits the binomial pattern, allowing us to use the theorem.

• First, identify each part of the expression: \( a = \frac{1}{k} \) and \( b = -\sqrt{3}p \).
• Then, note the exponent \( n = 3 \).
• For expansions, calculate each term separately using the binomial coefficients applicable for \( n = 3 \).

Through this methodical process, the expression can be expanded into a polynomial of terms as seen in the solution.

Polynomial Expansion

Polynomial expansion refers to expressing a polynomial in an expanded form, where all terms are explicitly written out. In the context of binomial expressions, such expansions often leverage the Binomial Theorem. By expanding \( \left( \frac{1}{k} - \sqrt{3}p \right)^3 \), we translate the compact expression into a fully expanded polynomial.
This process involves:

• Using binomial coefficients which are extracted using combination formulas \( \binom{3}{r} \).
• Calculating each specific term by adhering to the powers decremented for \( a \) and incremented for \( b \) from zero to the exponent \( n \).
• Bringing all calculated terms together to represent the expanded form of the original expression."
After applying this procedure, check each term to ensure no steps are skipped or miscalculated. In doing so, what began as a simple expression evolves into a detailed polynomial that shows exactly how it's built.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like adding or multiplying). They form the foundation of algebra and include operations on numbers and symbols (or "terms").
In our exercise, the expression \( \left( \frac{1}{k} - \sqrt{3}p \right)^3 \) is an example of such an expression. It's constructed through:

• Fractions, as seen in \( \frac{1}{k} \), which shows division in an algebraic sense.
• Exponents, wherein the cubing (raising something to the 3rd power) shows how expressions grow through multiplicative repetition.
• Radical terms, such as \( \sqrt{3} \), demonstrating operations involving roots.

Understanding this gives a grasp on how terms interact within polynomial expansions. Each component, whether it's a fraction or a radical, plays a role in delivering the expression to its final expanded form. This breakdown of components ultimately aids in mastering more complex algebraic operations.