Problem 104 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{25 x^{2}-40 x y+16 y^{2}}{x^{2} y^{4}} \cdot \frac{x y^{4}}{25 x^{2}-16 y^{2}}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{y}{x(5x + 4y)} \).

Step 1: Analyze the Expression

The expression consists of two rational expressions being multiplied: \( \frac{25x^{2}-40xy+16y^{2}}{x^{2}y^{4}} \) and \( \frac{xy^{4}}{25x^{2}-16y^{2}} \). To multiply these, focus on simplifying each part of the expression individually before performing the multiplication.

Step 2: Factor the Numerators

Factor the numerators. For \(25x^2 - 40xy + 16y^2\), use the quadratic form. It can be rearranged and factored as \((5x - 4y)^2\). The second numerator \( xy^4 \) does not need factoring.

Step 3: Factor the Denominators

Factor the denominators. Begin with \(x^2y^4\), which is already in its simplest form. For \(25x^2 - 16y^2\), recognize it as a difference of squares, \((5x + 4y)(5x - 4y)\).

Step 4: Perform the Multiplication

Write the full expression: \[\frac{(5x - 4y)^2 \cdot xy^4}{x^2y^4 \cdot (5x + 4y)(5x - 4y)}\]Multiply across the numerators and denominators.

Step 5: Simplify the Expression

Look for terms you can cancel out in the numerator and the denominator. The terms \((5x - 4y)\) appear in both and can be canceled. Additionally, \(xy^4\) cancels when divided by itself in the denominator.

Step 6: Simplify the Remaining Terms

After canceling, the expression simplifies to: \[\frac{xy}{x^2(5x + 4y)}\]This can simplify further to: \[\frac{y}{x(5x + 4y)}\] by canceling one \(x\) from the numerator and one from the denominator.

Step 7: Final Simplified Form

The final simplified expression is: \[\frac{y}{x(5x + 4y)}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is a key skill when dealing with algebraic expressions and especially important in simplifying rational expressions. When we factor polynomials, we essentially break them down into simpler, more fundamental elements that when multiplied together, give the original polynomial back. This process can be compared to breaking down numbers into their prime factors.

• **Quadratic Form**: Recognize the standard quadratic form, which is expressed as \( ax^2 + bx + c \). For example, the expression \( 25x^2 - 40xy + 16y^2 \) can be rearranged and identified as a perfect square trinomial. This specific form allows us to factor it into \((5x - 4y)^2\), revealing it as a repeated factor.
• **Difference of Squares**: This is another common pattern where an expression like \( a^2 - b^2 \) can be factored into \((a + b)(a - b)\). Applying this pattern to \( 25x^2 - 16y^2 \), we factor it as \((5x + 4y)(5x - 4y)\).

Understanding these patterns not only simplifies your work but also provides insight into the structure of the polynomial itself.

Multiplying Rational Expressions

Multiplying rational expressions involves combining two or more fractions into a single fraction. Just like when multiplying fractions numerically, you multiply the numerators together and the denominators together.
Here's how the process unfolds:

• **Multiply Across**: First, simply multiply across the numerators and across the denominators. For the expressions \(\frac{25x^2-40xy+16y^2}{x^2y^4} \) and \(\frac{xy^4}{25x^2-16y^2}\), this results in:\[ \frac{(5x - 4y)^2 \cdot xy^4}{x^2y^4 \cdot (5x + 4y)(5x - 4y)} \]
• **Simplify**: After multiplying, the key is to simplify. Look for common factors in the numerator and the denominator that can be cancelled out to reduce the expression to its simplest form.Keeping a checklist of patterns like the difference of squares and recognizing perfect squares allows you to factor and cancel efficiently.

This process highlights the importance of factoring, as it makes finding common factors that can be cancelled much easier.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the art of making an expression as straightforward as possible without changing its value. This involves reducing fractions, cancelling common terms in the numerator and denominator, and ensuring everything is expressed in the simplest form.
When you simplify the expression:

• **Cancel Common Factors**: Look for instances where terms in the numerator and denominator match. In our example, \((5x - 4y)\) cancels out, leaving fewer terms to work with. Similarly, \(xy^4\) cancels as it is present in both the numerator and denominator.
• **Reduce Further**: Always check if further reduction is possible. For instance, from \(\frac{xy}{x^2(5x + 4y)}\), you can further simplify by cancelling an \(x\) present in both the numerator and denominator, resulting in \(\frac{y}{x(5x + 4y)}\).

Through these steps, the original complex expression becomes manageable, and you gain a clearer understanding of its fundamental form.