Problem 84 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$27 y^{3}-1$$

Step-by-Step Solution

Answer

So, the factored form of \(27y^3 - 1\) is \((3y - 1)(9y^2 + 3y + 1)\).

Step 1: Identify the Values for 'a' and 'b'

The expression given is \(27y^3 - 1\). This expression should be perceived as a cubic minus a cubic, which means the difference of two cubes. Here, \(27y^3\) is the first cube and \(1\) is the second one. Hence, 'a' would be the cube root of \(27y^3\) which is \(3y\), and 'b' would be the cube root of \(1\) which is \(1\).

Step 2: Apply the Difference of Cubes Formula

The formula for the difference of two cubes is: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Substituting the values of 'a' as \(3y\) and 'b' as \(1\), into this formula gives: \((3y - 1)((3y)^2 + (3y)(1) + 1^2)\).

Step 3: Simplify the Factored Form

The factored form can be simplified as follows: \((3y - 1)(9y^2 + 3y + 1)\).

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial function represented by the general formula: \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a \) is non-zero. These functions are characterized by their three-dimensional shape, which has one or two turns and can exhibit behavior such as increasing and decreasing intervals, local maxima or minima, and inflection points. The graph of a cubic function is called a cubic parabola and has a unique property of having either one real root or three real roots when set equal to zero. This versatile behavior makes cubic functions pivotal in problems involving rate changes, such as velocity in physics.

Understanding cubic functions is essential while factoring cubic expressions, particularly when dealing with the sum or difference of two cubes. For example, in the expression \( 27y^3 - 1 \), the term \( 27y^3 \) represents a perfect cube that can be associated with a cubic function where \( a = 27, b = c = 0 \) and \( d = -1 \).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division, excluding division by a variable. Often, they're simplified or manipulated to solve equations or to represent various quantities. They don’t have an equality sign, unlike equations, which means they don’t generally have a 'solution' per se—rather, they are expressions that can be simplified or factored.

Keeping the components of algebraic expressions in mind is crucial while factoring expressions such as \( 27y^3 - 1 \). Recognizing that \( 27y^3 \) and \( 1 \) are each cubes of simpler expressions \( (3y)^3 \) and \( 1^3 \) respectively is an important step in the factoring process. This requires a good understanding of how algebraic expressions can be rewritten and manipulated, which is a fundamental skill in algebra.

Polynomial Factoring

Polynomial factoring is the process of breaking down a complex polynomial into simpler, 'factored' parts that, when multiplied together, give the original polynomial. It is an invaluable tool in solving polynomial equations because these factored forms can reveal the roots or solutions to the equation. There are different factoring techniques for different types of polynomials, such as extracting common factors, factoring by grouping, and special factoring formulas such as the difference of squares and the sum or difference of cubes.

In the context of the difference of two cubes \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), understanding and memorizing this formula allows students to factor expressions like \( 27y^3 - 1 \) effortlessly. Once \( a \) and \( b \) are identified as \( 3y \) and \( 1 \) respectively, the formula can be applied, ultimately simplifying the problem to \( (3y - 1)(9y^2 + 3y + 1) \), which helps as a step towards solving cubic equations or finding roots of cubic functions.