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Chapter 7: Problem 118

Perform the operations. Simplify, if possible \(\frac{x^{3}-y^{3}}{x^{3}+y^{3}} \div \frac{x^{3}+x^{2} y+x y^{2}}{x^{2} y-x y^{2}+y^{3}}\)

Short Answer

Expert verified
After simplification, the expression becomes \( \frac{(x-y)(x^2 y - xy^2 + y^3)}{(x+y)x} \).

Step by step solution

01

Identify Division of Fractions as a Multiplication

Recognize that dividing by a fraction is the same as multiplying by its reciprocal. Thus, rewrite the expression: \( \frac{x^3-y^3}{x^3+y^3} \cdot \frac{x^2 y - x y^2 + y^3}{x^3 + x^2 y + x y^2} \).
02

Factor the Numerator and Denominator of Each Fraction

Apply factorization formulas and factorize both the numerators and denominators. For the first fraction, \( x^3 - y^3 = (x-y)(x^2+xy+y^2) \) and \( x^3 + y^3 = (x+y)(x^2-xy+y^2) \).For the second fraction, notice: \( x^3 + x^2 y + x y^2 = x(x^2 + xy + y^2) \), and the numerator is already simplified.
03

Simplify the Expression

After substitution, the expression becomes: \( \frac{(x-y)(x^2+xy+y^2)}{(x+y)(x^2-xy+y^2)} \cdot \frac{x^2 y - x y^2 + y^3}{x(x^2 + xy + y^2)} \).Cancel common terms where applicable: \( (x^2+xy+y^2) \) cancels out from numerators and denominators in different fractions.
04

Multiply and Further Simplify

Multiply the simplified numerators and denominators:\( \frac{(x-y)(x^2 y - xy^2 + y^3)}{(x+y)x} \). If there's an opportunity for further simplification or factoring, make sure to do so, but here, it remains as it is.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Factorization
Polynomial factorization is crucial for simplifying expressions and finding common factors. It involves breaking down a complex polynomial into products of simpler polynomials. This can make operations like division and simplification much more manageable.
  • Firstly, observe the given expression: \( x^3 - y^3 \). Recognizing it as a difference of cubes is key. The formula for factoring a difference of cubes is: \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \).
  • Similarly, the sum of cubes, \( x^3 + y^3 \), can be factored as: \( (x+y)(x^2-xy+y^2) \), which helps simplify parts of the expression.
In the exercise, both the numerators and the denominators were expressed as cubes, making these formulas applicable to simplify the problem. This step aims to reduce complexity and prepare the expression for further operations.
Division of Fractions
When dividing fractions, it鈥檚 often more illustrative to think of it as multiplication by the reciprocal. This fundamental concept simplifies the operation and clears the path for further breakdown.
  • The division problem given as \( \frac{x^3-y^3}{x^3+y^3} \div \frac{x^3+x^2y+xy^2}{x^2y-xy^2+y^3} \) can be transformed by taking the reciprocal of the second fraction, changing the operation to a multiplication problem:
  • Rewriting it: \( \frac{x^3-y^3}{x^3+y^3} \cdot \frac{x^2y-xy^2+y^3}{x^3+x^2y+xy^2} \).
This reconstruction of the division is essential, as it allows you to use simple multiplication strategies, which can easily be coupled with factoring to reduce the expression further.
Expression Simplification
This is where the magic of all previous steps come together. Simplification aims to reduce expressions to their simplest form. This not only reveals the expression's true character but also makes it easier to understand.
  • After factorization and transforming division to multiplication, the expression is ripe for canceling out terms. For example, \( (x^2+xy+y^2) \) appears in both the numerator and denominator in different fractions, allowing for cancellation.
  • Upon canceling common terms, the operation converts into a simpler multiplication task: \( \frac{(x-y)(x^2y - xy^2 + y^3)}{(x+y)x} \). Though it seems complex, further simplification might not be possible as no more common terms are evident.
The ability to cancel terms and identify common factors is central to expression simplification. It's not just about reducing the number of terms, but understanding the structure of the expression itself.

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