Problem 68 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified result is \(x^3 - 9x^2 + 27x - 27\).

Step 1: Identify the Special Product Formula

The expression \((x-3)^3\) represents a cube of a binomial. We will use the special product formula for a binomial cube: \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Here, \(a = x\) and \(b = 3\).

Step 2: Apply the Formula to the Expression

Substitute \(a = x\) and \(b = 3\) into the cube of a binomial formula: \((x-3)^3 = x^3 - 3(x^2)(3) + 3(x)(3^2) - 3^3\).

Step 3: Perform Calculations on Each Term

Calculate each term separately:- \(x^3 = x^3\)- \(-3(x^2)(3) = -9x^2\)- \(3(x)(3^2) = 27x\)- \(-3^3 = -27\).Thus, the expanded form is \(x^3 - 9x^2 + 27x - 27\).

Step 4: Combine Results Into Final Answer

Put all the calculated terms together to get the simplified expression: \(x^3 - 9x^2 + 27x - 27\).

Key Concepts

Binomial Cube

When dealing with algebraic expressions, understanding special product formulas, like the binomial cube, is vital. A binomial is simply a polynomial with two terms, such as

• \((a + b)\)

The cube of a binomial takes one of these two-term expressions and raises it to the third power. An example of this is \((x - 3)^3\). With this action, we're essentially multiplying the binomial with itself three times. But, rather than expand everything manually, there's a useful formula to apply:

\((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). This formula simplifies the process significantly. In this exercise, the binomial being worked with is \((x-3)\), meaning \(a = x\) and \(b = 3\). Using this specific formula saves a great deal of time and simplifies complex calculations, making it easier to handle algebraic expressions more efficiently.

Algebraic Expressions

Algebraic expressions are a fundamental building block in mathematics. They involve combinations of numbers, variables, and operations. In the expression \((x-3)^3\), we have a simple algebraic expression composed of a variable \(x\) and a constant \(3\).

• Variables are symbols that represent numbers, typically changing based on given conditions or equations. Here, \(x\) can be any number.
• Constants are fixed numbers, like the 3 in our binomial.

Through algebraic operations such as addition, subtraction, multiplication, and exponentiation, complex ideas, and relationships can be expressed and solved. By applying special product formulas like the binomial cube, understanding and manipulating such expressions becomes more manageable. This is essential when solving real-world problems or performing more advanced calculations.

Simplification

Simplification is an important concept in algebra that aims to make expressions easier to understand and solve. It involves reducing expressions to their simplest form.

In the exercise given, after expanding the binomial cube \((x-3)^3\) using the special product formula, we simplify it to the expression \(x^3 - 9x^2 + 27x - 27\). This involves performing calculations to combine like terms, if any, and ensure the expression reflects its most reduced form.

• Combine terms: Look for terms that can be added or subtracted.
• Be precise: Ensuring each step in the simplification maintains the equality of the expression.

The end result is a more straightforward expression that is far easier to comprehend and work with, both theoretically and practically. Simplification thus makes it feasible to identify relationships and insights that would otherwise be hidden in more complicated expressions.