Problem 82 - Elementary and Intermediate Algebra

Question

Factor. $$ x^{3 m}-y^{3 n} $$

Step-by-Step Solution

Answer

The expression is factored as \((x^m - y^n)(x^{2m} + x^m y^n + y^{2n})\).

Step 1: Identify the Expression Format

The expression given is \( x^{3m} - y^{3n} \), which is a difference of cubes. A difference of cubes follows the formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \).

Step 2: Assign Appropriate Values

From \( x^{3m} - y^{3n} \), assign \( a = x^m \) and \( b = y^n \). This makes the provided expression equal to \( (x^m)^3 - (y^n)^3 \), fitting the difference of cubes formula.

Step 3: Apply the Difference of Cubes Formula

Substitute \( a = x^m \) and \( b = y^n \) into the difference of cubes formula. This gives:\[(x^m - y^n)( (x^m)^2 + x^m y^n + (y^n)^2 )\].

Step 4: Simplify Each Factor

Simplify each part of the expression:The first factor is \( (x^m - y^n) \).The second factor is:- \((x^m)^2 = x^{2m}\)- \(x^m y^n\) remains as is- \((y^n)^2 = y^{2n}\)Thus, the second factor becomes: \( x^{2m} + x^m y^n + y^{2n} \).

Step 5: Write the Final Factored Expression

Combine both factors to obtain the factored form:\[(x^m - y^n)(x^{2m} + x^m y^n + y^{2n})\].

Key Concepts

Difference of Cubes

In algebra, one of the most important factorization techniques is the difference of cubes. This method is useful when you want to factor an expression of the form \(a^3 - b^3\). When you encounter such an expression, you can use the formula: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\] This formula simplifies the process of breaking down the expression into a product of simpler terms.

• The first factor is \(a - b\), where \(a\) and \(b\) are the cube roots of the terms.
• The second factor is \(a^2 + ab + b^2\), which is a bit more complex but still manageable.

Applying this formula lets us express complex cubic expressions as a product of polynomials, making them easier to solve or simplify further.

Exponent Rules

Exponents are a way to express repeated multiplication of a base number. For example, \(x^3\) means \(x\) multiplied by itself three times. In the context of our expression \(x^{3m} - y^{3n}\), we deal with exponents in several steps:When raising an exponent to another exponent, you multiply the exponents according to the rule \((x^m)^3 = x^{m \times 3} = x^{3m}\). This simplifies calculations significantly:

• For \((x^m)^2\), the result is \(x^{2m}\).
• For \((y^n)^2\), it becomes \(y^{2n}\).

Exponent rules are crucial in algebra as they help simplify expressions, allowing easier factoring, solving, and graphing.

Algebraic Formulas

Algebraic formulas like the difference of cubes come in handy when working with polynomial expressions. Formulas in algebra are often designed as shortcuts to simplify complex operations into manageable steps. The difference of cubes formula we applied is a textbook example of using algebraic formulas to rearrange terms efficiently:

• Structure Understanding: Recognizing \(a^3\) and \(b^3\) helps in identifying the type of expression we are dealing with.
• Simplification: An algebraic formula provides a streamlined path to reduce the expression into simpler, factorable parts.
• Application: Once the expression's type is recognized, formulas allow easy substitution of terms aligned with the given structure.

Familiarity with various algebraic formulas is vital, as it transforms complex factoring tasks into routine operations that are easy to manage. This understanding not only helps in calculations but also in portraying the mathematical relationships clearly.