Problem 43 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Special Product

The expression \((1 - 2r)^3\) is a cube of a binomial. This fits the structure of the binomial cube expansion, which is given by the formula \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Here, \(a = 1\) and \(b = 2r\).

Step 2: Apply the Binomial Cube Formula

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

Step 3: Simplify Each Term

Calculate each term separately: - \(1^3 = 1\), - \(-3 \cdot 1^2 \cdot 2r = -6r\), - \(3 \cdot 1 \cdot (2r)^2 = 3 \cdot 4r^2 = 12r^2\), - \(-(2r)^3 = -8r^3\).

Step 4: Combine Like Terms

Add the simplified terms together: \[1 - 6r + 12r^2 - 8r^3\]. There are no like terms to combine further, so this is the final simplified expression.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra for expanding expressions that are raised to a power. It provides a formula for expanding binomials, expressions with two terms, into polynomials. The general form for the Binomial Theorem is:

• For any integer \( n \), \((a + b)^n\) expands to \(\sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\).

Here, \( {n \choose k} \) represents the binomial coefficient, which is calculated as \(\frac{n!}{k!(n-k)!}\).
This theorem simplifies the process of raising binomials to high powers, saving time over directly multiplying out terms.
In simpler terms, understanding the binomial expansion can help break down more complex expressions into manageable parts. This way, each component can be tackled more straightforwardly, as seen in specific formulas like the cube of a binomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In algebra, we use these expressions to represent mathematical relationships. For example,

• An expression like \(1 - 2r\) involves a constant \(1\) and a variable \(r\).
• Multiplication and powers can be applied to these expressions, such as raising \((1-2r)\) to the power of 3.

They serve as building blocks for forming equations and functions.
Algebraic expressions often include terms that may need simplification. For example, when applying the binomial theorem or other special product formulas, each term should be simplified as illustrated in the cube expansion of a binomial. Understanding these expressions can aid in performing operations like addition, subtraction, multiplication, and division on equations, laying the groundwork for solving algebraic problems.

Polynomial Expansion

Polynomial expansion is the process of multiplying out expressions and simplifying them into a polynomial form. A polynomial is an expression consisting of variables, coefficients, and exponents, connected by addition, subtraction, and multiplication.

• Using special product formulas, like the one applied in our example of \((1 - 2r)^3\), helps simplify the expansion process.
• The expansion converts expressions into a sum of individual terms, resulting in a polynomial.

Each polynomial term is then simplified, often by combining like terms.
For instance, in the expanded form of \((1-2r)^3\), you end up with \(1 - 6r + 12r^2 - 8r^3\), a polynomial with four distinct terms.
Mastering polynomial expansion is crucial, as it is widely used in various areas of mathematics. It makes working with complex expressions more manageable by breaking them down into components that are easier to handle and understand.