Problem 9 - Algebra 1: Concepts and Skills

Question

Factor the expression. \(27 x^{3}+1\)

Step-by-Step Solution

Answer

\((3x+1)(9x^{2}-3x+1)\)

Step 1: Identify the Cubes

We first identify the cubic terms in the given expression. In the expression \(27x^{3}+1\), \(27x^{3}\) is the cube of \(3x\) and 1 is the cube of 1.

Step 2: Apply the Sum of Cubes Formula

We now apply the formula for factoring the sum of cubes \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Substituting \(a=3x\) and \(b=1\) into the formula, we get, \((3x + 1)((3x)^{2} - 3x*1 + 1^{2})\)

Step 3: Simplify the Expression

The final step is to simplify the expression we obtained in Step 2. When we simplify, we get \((3x+1)(9x^{2}-3x+1)\).

Key Concepts

Cubic expressions

Cubic expressions are polynomials where the highest degree of any term is three, providing them with distinctive characteristics. In a general cubic expression, such as \(ax^3 + bx^2 + cx + d\), the term \(ax^3\) is identified as the cubic term. The coefficient \(a\) can never be zero, as it dictates the polynomial’s cubic nature.
The expression we are focusing on is \(27x^3 + 1\). Here, \(27x^3\) is the cubic term. Recognizing these elements in a cubic expression is essential for further algebraic manipulations.

Sum of cubes

The sum of cubes is a special category of polynomial expressions that can be factored using a fixed formula. This applies to expressions structured as \(a^3 + b^3\).
The formula for factoring a sum of cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). This method simplifies complex expressions into manageable components. It’s particularly handy when working with expressions that superficially seem intricate.
In our exercise, \(27x^3 + 1\), we identified \(a = 3x\) and \(b = 1\). Applying the sum of cubes formula converts the expression into a product: \((3x + 1)(9x^2 - 3x + 1)\). This step-by-step approach makes it easier to handle and solves the problem efficiently.

Polynomial equations

A polynomial equation is a mathematical expression consisting of variables and coefficients. These expressions can have various degrees depending on the highest power of the variable present. They take the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\).
Cubic polynomial equations, such as \(x^3 - 3x + 1 = 0\), are a subset characterized by their degree of three. They can have three roots, which might be real or complex numbers.
In the context of the current problem, factoring the polynomial \(27x^3 + 1\) into \((3x + 1)(9x^2 - 3x + 1)\) allows us to potentially solve related polynomial equations. Understanding how to factor these expressions yields insights into their roots and helps in simplifying computations in algebra.