Problem 101 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(10 m^{9}-270 n^{6}\)

Step-by-Step Solution

Answer

The factored form is \(10(m^3 - 3n^2)(m^6 + 3m^3n^2 + 9n^4)\).

Step 1: Identify the Greatest Common Factor

First, identify the greatest common factor (GCF) of the two terms. Both terms have coefficients, 10 and 270, and no common variables. The largest number that can divide both 10 and 270 is 10. Factor 10 out of the expression.

Step 2: Factor Out the GCF

Now, factor 10 out of each term:\[10m^9 - 270n^6 = 10(m^9 - 27n^6)\]

Step 3: Recognize the Form

Notice the expression inside the parentheses, \(m^9 - 27n^6\), is a difference of cubes. It can be rewritten as \((m^3)^3 - (3n^2)^3\) since \(27n^6 = (3n^2)^3\).

Step 4: Apply the Difference of Cubes Formula

Use the difference of cubes formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = m^3\) and \(b = 3n^2\). Substitute to get:\[(m^3 - 3n^2)((m^3)^2 + m^3(3n^2) + (3n^2)^2)\]

Step 5: Simplify Further

Simplify the expression in the parentheses:- Calculate \((m^3)^2 = m^6\)- Calculate \(m^3(3n^2) = 3m^3n^2\)- Calculate \((3n^2)^2 = 9n^4\)Thus, the resulting expression is:\[(m^3 - 3n^2)(m^6 + 3m^3n^2 + 9n^4)\]

Step 6: Construct the Final Factored Form

Combine all the factors including the GCF:\[10(m^3 - 3n^2)(m^6 + 3m^3n^2 + 9n^4)\]This is the fully factored form of the original expression.

Key Concepts

Understanding the Greatest Common Factor (GCF)

When working with polynomials, finding the greatest common factor, or GCF, is always a good starting point. The GCF is the largest factor that both terms in the expression share equally. In our example, the expression is a binomial: \(10m^9 - 270n^6\). To determine the GCF, we look at the coefficients, which are 10 and 270. Since the largest number that divides both 10 and 270 is 10, this becomes our GCF. Factoring this out simplifies the expression and makes further operations, like identifying more intricate patterns, much easier.

By factoring out the GCF, we get: \[10(m^9 - 27n^6)\]

At this point, each term inside the parentheses is simpler and has a more straightforward mathematical pattern, which is a valuable step in factoring more complex polynomials. Always remember that extracting the GCF is essential, not only for simplifying expressions but also for paving the way to recognize other algebraic structures.

The Difference of Cubes

The difference of cubes is a unique tool in algebra for simplifying expressions. Whenever you see a term like \(a^3 - b^3\), it tells us the inner structure is a "difference of cubes." This formula is especially useful when the expressed cubes are not immediately obvious.

In our case, the middle term \(m^9 - 27n^6\) can be expressed as cubes: \((m^3)^3 - (3n^2)^3\). Recognizing this starts from identifying perfect cube terms, which are powers of three.

Using the "difference of cubes" formula:

• Formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).
• Set \(a = m^3\) and \(b = 3n^2\).

This breakdown helps in easily rewriting the expression to: \[(m^3 - 3n^2)(m^6 + 3m^3n^2 + 9n^4)\].

Understanding this shortcut saves time and effort, helping solve polynomial equations efficiently. This methodology offers a structured mechanism for handling complex cubic polynomials.

Algebraic Approaches to Polynomial Factoring

Algebra isn't just about solving equations—it's about understanding connections between numbers and symbols. Factoring polynomials, such as \(10m^9 - 270n^6\), requires recognizing patterns and systematic strategies.

Each polynomial has its distinct structure. Breaking them down into smaller, manageable parts simplifies the equation.

First, using the greatest common factor simplifies the polynomial. Then, identifying algebraic patterns such as sums or differences of cubes, or even squares, allows a deeper understanding of the polynomial.

Common algebraic techniques include:

• Recognizing patterns: spotting binomials, trinomials, and using formulaic expressions like difference of squares or cubes.
• Simplifying through common factors, making complex expressions more solvable.

By learning and applying these techniques, one becomes adept at polynomial operations, enhancing confidence and problem-solving skills in algebra. Algebra then transforms into more than just a set of rules, becoming a powerful tool for interpreting and interacting with mathematical expressions.