Chapter 4: Problem 58
Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ 8 m^3-343 $$
Short Answer
Expert verified
\((2m - 7)(4m^2 +14m + 49)\
Step by step solution
01
Identify the Expression
First, recognize that \(8m^3 - 343\) is a difference of cubes. It means it can be written in the form \(a^3 - b^3\), where \(a\) and \(b\) are perfect cubes.
02
Rewrite the Expression
For \(8m^3 = (2m)^3\) and \(343 = 7^3\), the expression can be rewritten as \((2m)^3 - 7^3\). This is our denominator in the format \(a^3 - b^3\).
03
Apply the Formula for Difference of Cubes
Apply the formula for the difference of cubes, which is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). For our values of \(a\) and \(b\) as \(2m\) and \(7\) respectively, plug them into the formula. The factored form becomes \((2m - 7)[(2m)^2 + (2m)(7) +7^2]\).
04
Simplify the Equation
Lastly, simplify the bracketed term to obtain the fully factored expression as \((2m - 7)(4m^2 +14m + 49)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
difference of cubes
A difference of cubes occurs when you have a polynomial in the form of \(a^3 - b^3\). This format indicates that both terms are perfect cubes. Recognizing this pattern is crucial for effective factoring. Let's break down why we use the difference of cubes approach here:
- First, identify each term as a cube. For example, in the expression \(8m^3 - 343\), \(8m^3\) is a cube of \((2m)\) and \(343\) is a cube of \(7\).
- The polynomial can be rewritten as \((2m)^3 - 7^3\), which matches \(a^3 - b^3\).
- This form allows us to use a specific factoring formula designed for cubes.
factoring techniques
Factoring techniques are strategies used to rewrite polynomials as products of simpler terms. One essential technique involves recognizing patterns like the difference of cubes. Here are steps you can follow to factor a polynomial using the difference of cubes:
- Recognize the Pattern: Identify whether the polynomial is a perfect cube difference. This involves ensuring each term is expressed as a cube.
- Express in Cube Form: Convert each term into its respective cube form, such as \((2m)^3\) and \(7^3\) in our example of \(8m^3 - 343\).
- Apply the Formula: Use the difference of cubes formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Substituting \(a = 2m\) and \(b = 7\) transforms the polynomial into simpler factors.
- Simplify: Finally, simplify the expression inside the brackets to complete the factoring process.
polynomial expressions
Polynomial expressions consist of variables raised to any whole-number power and their coefficients. Understanding how to manipulate these expressions is foundational in algebra. In the case of \(8m^3 - 343\), the polynomial initially appears as a complex expression.
- Structure of the Polynomial: The given polynomial \(8m^3 - 343\) is made up of two terms: one involving a variable \(m\), raised to the third power, and a constant.
- Identifying Cubes: Recognizing that both terms represent cubes facilitates the use of advanced factoring approaches like the difference of cubes. This recognition turns the polynomial from a single expression into the product of several factors.
- Simplifying Expressions: By factoring, we transform the original polynomial into a product of linear and quadratic factors, \((2m - 7)(4m^2 + 14m + 49)\). This reveals more about the polynomial's nature, like its roots and behavior.