Problem 42 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(x-3)^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{3} - 9x^{2} + 27x - 27\).

Step 1: Identify the Special Product Formula

The expression \((x - 3)^{3}\) follows the cube formula for binomials. The formula for cubing a binomial \((a - b)^{3}\) is given by \(a^{3} - 3a^{2}b + 3ab^{2} - b^{3}\). Here, identify \(a = x\) and \(b = 3\).

Step 2: Substitute Values into the Formula

Substitute the identified values \(a = x\) and \(b = 3\) into the special product formula: \((x - 3)^{3} = x^{3} - 3x^{2}(3) + 3x(3)^{2} - (3)^{3}\).

Step 3: Perform the Calculations

Calculate each of the terms: 1. \(x^{3}\) is equal to \(x^{3}\).2. \(3x^{2}(3)\) is equal to \(9x^{2}\).3. \(3x(3)^{2}\) is equal to \(27x\).4. \((3)^{3}\) is equal to \(27\).

Step 4: Combine the Calculated Terms

Substitute the calculations back into the expression: \(x^{3} - 9x^{2} + 27x - 27\). This is the simplified form of \((x-3)^{3}\).

Key Concepts

Binomial Expansion

Binomial expansion is a method used to express a binomial raised to a power, like \((x-3)^3\), in an expanded form. When expanding binomials, we often utilize special product formulas to simplify the process. One particularly useful formula is the binomial theorem, which provides a structured approach to expanding powers of binomials.

For binomials of the form \((a-b)^n\), where \(n\) is a whole number, a special pattern emerges. The expansion follows a predictable sequence involving powers of \(a\) and \(b\), and coefficients that correspond to binomial coefficients. These coefficients can be found using Pascal’s Triangle, or calculated using combinations (\(\binom{n}{k}\)).

Using a special product formula for cubing, such as \((a-b)^3=a^3 - 3a^2b+3ab^2-b^3\), greatly simplifies the calculation process. By substituting specific values into this formula, it becomes straightforward to write the expanded form.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operation symbols. In the context of binomial expansion, these expressions consist of terms involving powers of variables and coefficients. Understanding these expressions is crucial for performing operations like multiplication and simplification.

In the example, \((x-3)^3\), is an algebraic expression consisting of a binomial \((x-3)\). Each application of a special product formula involves evaluating expressions to arrive at expanded and simplified forms. The terms within this expression, such as \(x\) and \(-3\), play important roles in determining the structure of the final polynomial.

When working with algebraic expressions, it is helpful to identify the indeterminate, here \(x\), and the constant, \(-3\). Such identification aids in substituting and manipulating the expression, ensuring accurate simplification and expansion results.

Polynomial Simplification

Polynomial simplification involves combining like terms and reducing expressions to their simplest form. This process is crucial in obtaining a manageable expression from a more complex form.

Once an expression is expanded, like from \((x-3)^3\) to \(x^3 - 9x^2 + 27x - 27\), simplification focuses on ensuring that all similar terms are aggregated, and coefficients are simplified appropriately. This allows the polynomial to present the most concise and clear form possible.

Simplification may also involve rearranging terms to follow a particular order, usually descending powers, which helps in easily interpreting the polynomial. By factoring and re-evaluating components if necessary, one can assure that the polynomial adequately represents the initial algebraic or arithmetic problem.