Problem 64 - Elementary Algebra

Question

Factor completely. $$ 343 m 3-1 $$

Step-by-Step Solution

Answer

The expression factors to \((7m-1)(49m^2+7m+1)\).

Step 1: Identify the Expression

The given expression is: \[ 343m^3 - 1 \] which can be seen as a difference of cubes because both 343 and 1 are perfect cubes.

Step 2: Express as a Difference of Cubes

Recognize that the expression \( 343m^3 - 1 \) can be rewritten as a difference of cubes:\[ a^3 - b^3 \] where \( a = 7m \) and \( b = 1 \). This is because \( 343 = 7^3 \) and \( 1 = 1^3 \).

Step 3: Apply Difference of Cubes Formula

The difference of cubes formula is:\[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]Substituting \( a = 7m \) and \( b = 1 \) into the formula gives:\[ (7m - 1)((7m)^2 + (7m)(1) + (1)^2) \]

Step 4: Simplify Each Part

Simplify the expression:1. First term: \( 7m - 1 \)2. Second term: Calculate \((7m)^2 = 49m^2\)3. Middle term: Calculate \(7m \times 1 = 7m\)4. Last term: Calculate \(1^2 = 1\)Putting it all together:\[ 49m^2 + 7m + 1 \]

Step 5: Write Final Factored Form

The completely factored form of the expression is:\[ (7m - 1)(49m^2 + 7m + 1) \]

Key Concepts

Difference of Cubes

The concept of the \(\textit{difference of cubes}\) refers to a specific type of algebraic expression where one term is subtracted from another, and each term is a cube. For example, in an expression like \( a^3 - b^3 \), both terms are cubes of \( a \) and \( b \) respectively.
To break this down, let's consider the expression \( 343m^3 - 1 \). Here, each component, \( 343m^3 \) and \( 1 \), is a perfect cube. Specifically, \( 343 = 7^3 \) and \( 1 = 1^3 \), allowing us to rewrite the expression as \( (7m)^3 - 1^3 \).
This equation fits perfectly into the difference of cubes structure, which is crucial because it allows us to apply a special factoring formula to simplify it. Recognizing and rewriting expressions as a difference of cubes is an essential skill in algebra.

Polynomial Expressions

\(\textit{Polynomial expressions}\) are mathematical expressions consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication. These expressions can be complex and involve various powers of variables.
In the expression \( 343m^3 - 1 \), we are dealing with a cubic polynomial because the highest power of the variable \( m \) is 3. Cubic polynomials like this one can often be rewritten using algebraic identities, making them easier to factor.
Understanding polynomial expressions involves identifying their components, such as the coefficients (343 and 1 in this case), the variable (\( m \)), and the degree of the polynomial (3). This knowledge is foundational when manipulating and simplifying algebraic expressions.

Algebraic Factoring Methods

\(\textit{Algebraic factoring methods}\) are techniques used to rewrite polynomials as products of simpler expressions. One specific method is factoring via the difference of cubes formula, which is employed when an expression takes the form \( a^3 - b^3 \).
The difference of cubes is factored using the formula: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).
Using our earlier example, substitute \( a = 7m \) and \( b = 1 \) into the formula:

• First term: \( 7m - 1 \)
• Square term: \( (7m)^2 = 49m^2 \)
• Middle term: \( 7m \times 1 = 7m \)
• Last term: \( 1^2 = 1 \)

Putting it all together, the expression becomes \( (7m - 1)(49m^2 + 7m + 1) \). This complete factorization showcases how powerful factoring techniques can simplify even seemingly complex polynomial expressions.