Problem 24 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{x^{2}+y^{2}}{x-y} \cdot \frac{x^{2}-x y}{3}$$

Step-by-Step Solution

Answer

Multiplication of fractions results in a form that appears simplified with current constraints.

Step 1: Analyze the Given Expression

The given expression is \( \frac{x^{2}+y^{2}}{x-y} \cdot \frac{x^{2}-xy}{3} \). We need to multiply these two fractions together and simplify the result.

Step 2: Multiply the Numerators and Denominators

When multiplying fractions, multiply the numerators together and the denominators together. The resulting expression is:\[ \frac{(x^2 + y^2) \cdot (x^2 - xy)}{(x - y) \cdot 3} \]

Step 3: Expand the Numerator

Now, expand the product in the numerator: \( (x^2 + y^2)(x^2 - xy) = x^2(x^2 - xy) + y^2(x^2 - xy) \). This gives us:\[ x^4 - x^3y + x^2y^2 - x^2y^3 \]

Step 4: Verify a Simple Factorization

Check if there's a simpler factorization or cancellation in the numerator and denominator:Currently, there are no obvious simplifications between the expanded form \( x^4 - x^3y + x^2y^2 - x^2y^3 \) and the denominator \( 3(x - y) \). We need to re-evaluate our current simplicity.

Step 5: Re-assess Any Factoring Errors

Upon re-evaluating, notice that simplifying directly through complex factoring might lead nowhere due to non-trivial polynomial identities without more specific information.

Step 6: Conclude on Simplicity

Given these steps and seeing no viable simpler form without arbitrary assumptions, the expression \( \frac{x^2 + y^2}{x-y} \cdot \frac{x^2 - xy}{3} \) stands as simplified as possible given the constraints presented and typical school-level algebra techniques.

Key Concepts

Simplifying Algebraic Expressions

When working with algebraic expressions, simplification is crucial to make complex problems more manageable. Simplifying involves reducing the expression to its simplest form without changing its value. Key techniques include:

• Combining like terms, where we sum or subtract coefficients of similar terms, like \(3x + 2x = 5x\).
• Using distributive property, such as transforming \(a(b+c)\) into \(ab + ac\).

In the given exercise, simplification involves attempting to factor or cancel terms in the product of fractions. Factors, on both numerator and denominator sides, are analyzed for potential reduction. However, factors in non-trivial polynomials may not always present an obvious path to simplification without additional context, as in the example \(x^4 - x^3y + x^2y^2 - x^2y^3\). Ambiguous simplifications may occur, signaling the necessity for careful examination of every step.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in the other polynomial. This process follows the distributive property principle, making sure every combination of terms is accounted for. The steps include:

• First, rewrite the polynomials so each is expanded, showing visible terms.
• Multiply each term in the first polynomial by every term in the second.
• Combine like terms to simplify further if necessary.

In our example, \((x^2 + y^2)(x^2 - xy)\) exemplifies polynomial multiplication. This required multiplying \(x^2\) by every term in the second expression \(x^2 - xy\), and similarly for \(y^2\). Eventually, this gives us polynomials as shown: \[x^4 - x^3y + x^2y^2 - x^2y^3\]. It is crucial, though, to simplify by combining terms or factoring, but that step depends, as mentioned, on specific problem constraints.

Fraction Multiplication Techniques

Multiplying fractions, whether numeric or algebraic, involves multiplying the numerators and denominators separately. The general steps include:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Simplify if possible, by canceling shared factors.

Consider the expression \(\frac{x^2 + y^2}{x-y} \cdot \frac{x^2 - xy}{3}\). The key here is that the multiplication leads to a new expression: \(\frac{(x^2 + y^2)(x^2 - xy)}{3(x-y)}\). Analyzing for potential simplifications through factors across the fraction is typical, expecting any common terms to cancel. When common simplifications aren't evident, as in this exercise, it often concludes with recognizing the limits of simplification given the problem’s algebraic structure.