Problem 69 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((1-2 r)^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(1 - 6r + 12r^2 - 8r^3\).

Step 1: Identify the Special Product Formula

The expression \((1-2r)^3\) corresponds to a binomial raised to the third power, which can be expanded using the formula for the cube of a binomial, \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).

Step 2: Assign Values to the Formula Components

Identify \(a\) and \(b\) from the expression \((1-2r)^3\): here, \(a = 1\) and \(b = 2r\).

Step 3: Apply the Special Product Formula

Substitute \(a = 1\) and \(b = 2r\) into the formula: \[(1-2r)^3 = 1^3 - 3(1)^2(2r) + 3(1)(2r)^2 - (2r)^3\]

Step 4: Calculate Each Term Separately

Perform the calculations for each term: 1. \(1^3 = 1\)2. \(-3(1)^2(2r) = -6r\)3. \(3(1)(2r)^2 = 3(4r^2) = 12r^2\)4. \(-(2r)^3 = -8r^3\)

Step 5: Simplify the Expression

Combine all the calculated terms to simplify the expression:\[1 - 6r + 12r^2 - 8r^3\]

Key Concepts

Binomial Expansion

The concept of binomial expansion involves expressing a binomial raised to a power as a sum of terms. A binomial is an algebraic expression that contains two terms. When we raise a binomial to a power, we apply a specific formula designed to simplify the process of expansion.
For example, the binomial expansion of ewline

• ewlinea binomial squared is ewlineewline
  • ewlineewlineewlineewlineewline\((a + b)^2 = a^2 + 2ab + b^2\).ewline

The expansion progressively develops more terms with increasing powers, each term consisting of the products of the binomial terms raised to various powers.For higher powers like a binomial cubed, the terms become more numerous and involve higher powers of each individual component.

Cube of a Binomial

For a binomial raised to the power of three, we use what is often referred to as the cube of a binomial. This formula allows us to expand an expression such as \((a-b)^3\). The formula is structured as:

• \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).

Each of these terms originates from a systematic expansion. Notice the alternation of signs and how the powers of the terms systematically reduce and increase.
To apply this formula, identify the values for \(a\) and \(b\) in your binomial expression. Then directly substitute these values into the formula to achieve an expanded expression.
This formula simplifies the task of cubing a binomial significantly, turning what could be a daunting calculation into a straightforward process.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and involve combinations of variables, numbers, and operations. They are the building blocks of equations and are used to describe patterns, relationships, and real-world phenomena.
Within an algebraic expression:

• Variables represent unknown quantities and are typically denoted by letters.
• Constants are known, fixed numbers.
• Operations (such as addition, subtraction, multiplication) connect variables and constants to form an expression.

Understanding how to manipulate algebraic expressions is crucial for performing operations like expansion, simplification, and solving equations.

Polynomial Simplification

Polynomial simplification involves reducing an expression to its simplest form by combining like terms and performing arithmetic operations. When simplifying a polynomial, it is essential to recognize terms that can be combined and cancel out terms that oppose one another.
In the context of binomial expansion, after applying the special product formula, the expression often contains multiple terms. To simplify:

• Collect all like terms, which are terms with the same variables raised to the same power.
• Perform addition or subtraction as required to combine these terms.

For example, after expanding \((1-2r)^3\), you get an expression with terms that can often be further simplified into a neat polynomial form. Simplifying expressions makes them easier to read and use in further calculations or real-world applications.