Problem 29 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ \left(2 p^{3}-3\right)^{3} $$

Step-by-Step Solution

Answer

The expanded form is \(8p^9 - 36p^6 + 54p^3 - 27\).

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, \(a = 2p^3\), \(b = -3\), and \(n = 3\).

Step 2: Write Down the General Expression

Write \((2p^3 - 3)^3\) using the binomial theorem: \[(2p^3 - 3)^3 = \sum_{k=0}^{3} \binom{3}{k} (2p^3)^{3-k} (-3)^{k}\].

Step 3: Calculate Binomial Coefficients

The binomial coefficients are calculated as follows: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\).

Step 4: Expand the Binomial Expression

Substitute values and simplify each term:1. For \(k = 0\): \(\binom{3}{0} (2p^3)^{3-0} (-3)^0 = 1 \cdot (2p^3)^3 \cdot 1 = 8p^9\).2. For \(k = 1\): \(\binom{3}{1} (2p^3)^{3-1} (-3)^1 = 3 \cdot (2p^3)^2 \cdot (-3) = -36p^6\).3. For \(k = 2\): \(\binom{3}{2} (2p^3)^{3-2} (-3)^2 = 3 \cdot (2p^3)^1 \cdot 9 = 54p^3\).4. For \(k = 3\): \(\binom{3}{3} (2p^3)^{3-3} (-3)^3 = 1 \cdot 1 \cdot (-27) = -27\).

Step 5: Combine All Terms

Add all calculated terms together: \(8p^9 - 36p^6 + 54p^3 - 27\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. In the given exercise, the expression \((2p^3 - 3)^3\) is an example of an algebraic expression.
Since it consists of both a variable part \(2p^3\) and a constant part \(-3\), they are combined under exponentiation.
These expressions can be simplified, expanded, or evaluated for specific values using various mathematical techniques. Understanding algebraic expressions is crucial, as it forms the basis for manipulating more complex equations in mathematics.

Polynomial Expansion

Polynomial expansion is the process of expanding expressions that involve powers or products of polynomials.
The Binomial Theorem helps in expanding expressions of the form \((a + b)^n\) into a sum of individual terms.
This makes polynomial expansion more efficient and systematic. In our example, \((2p^3 - 3)^3\) is expanded using the theorem, giving each term as a separate entity that can be calculated individually.

• First, apply coefficients from Pascal's Triangle to determine the multiplicity of each term.
• Use powers of each component, \(a = 2p^3\) and \(b = -3\), to expand into individual terms.
• Finally, combine these terms into a simplified polynomial expression.

Polynomial expansion allows complex expressions to be broken down into comprehensible pieces.

Mathematical Theorems

Mathematical theorems provide rigorous relationships and rules that help solve problems more effectively. The Binomial Theorem is one such vital theorem in mathematics.
It states that for positive integers \(n\), an expression of the form \((a + b)^n\) can be expanded into a sum, where each term is calculated based on coefficients and powers of \(a\) and \(b\).
The coefficients are derived from the binomial coefficients, commonly seen in Pascal's Triangle.

• This theorem is essential for expanding polynomial expressions efficiently.
• It reduces the risk of errors by providing a structured path to follow.
• Understanding the Binomial Theorem opens doors to more advanced mathematical concepts, such as calculus or number theory.

These theorems form the backbone of mathematical logic, making complex problems manageable.