Problem 78 - Algebra A Combined Function

Question

Multiply or divide as indicated. $$ \left(\frac{x^{2} y^{2}-x y}{4 x-4 y} \div \frac{3 y-3 x}{8 x-8 y}\right) \cdot \frac{y-x}{8} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{xy(xy - 1)}{96} \).

Step 1: Rewrite the Expression

The given expression is \( \left(\frac{x^2 y^2 - xy}{4x - 4y} \div \frac{3y - 3x}{8x - 8y}\right) \cdot \frac{y - x}{8} \). Start by rewriting the division as multiplication by the reciprocal of the second fraction: \[ \frac{x^2 y^2 - xy}{4x - 4y} \times \frac{8x - 8y}{3y - 3x} \] which is then multiplied by \( \frac{y - x}{8} \).

Step 2: Factorization

Factor each term. The numerator \( x^2 y^2 - xy \) can be factored as \( xy(xy - 1) \). The denominator \( 4x - 4y \) can be factored as \( 4(x - y) \). Similarly, \( 3y - 3x = -3(x-y) \) and \( 8x - 8y = 8(x-y) \). The expression is then: \[ \frac{xy(xy - 1)}{4(x-y)} \times \frac{8(x-y)}{-3(x-y)} \times \frac{y-x}{8} \].

Step 3: Simplify the Expression

Cancel out common factors. \( (x-y) \) and \( 8 \) are common in the terms. The expression simplifies as: \[ \frac{xy(xy-1)}{4} \times \frac{1}{-3} \times \frac{-1}{8} \] becomes: \[ \frac{xy(xy-1)}{4 \times 3 \times 8} = \frac{xy(xy-1)}{96} \].

Step 4: Consolidate the Result

Combining the multiplications, the final expression is found by multiplying the factors: \[ \frac{xy(xy - 1)}{96} \].

Step 5: Final Step: Write Final Expression

The simplified result of the original expression is \[ \frac{xy(xy - 1)}{96} \]. This is the simplest form for the given problem.

Key Concepts

Factorization

Factorization is a crucial concept in algebra that involves breaking down a composite number, polynomial, or expression into a product of simpler factors. This can simplify expressions and make them easier to work with. In our original exercise, factorization is used to decompose polynomial expressions into their most basic components. For instance, the expression \(x^2y^2 - xy\) was factored into \(xy(xy - 1)\). This essentially means finding terms that multiply together to give the original expression. Another factorization example from our expression is the term \(4x - 4y\), which was rewritten as \(4(x - y)\).

To factor an expression:

• First, look for common factors in all terms. These are numbers or variables that evenly divide each term in the expression.
• Once you've found these, factor them out to simplify the equation.
• Continue to look for other factorization opportunities, such as spotting differences of squares or perfect square trinomials.

By working through these processes, complex expressions become easier to manage and solve when used in equations.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. In these expressions, operations like addition, subtraction, multiplication, and division align with those of regular fractions. The original exercise involved rational expressions where each part, both numerators and denominators, were represented by polynomial expressions.

Handling these requires steps similar to those for fractions:

• Simplify by canceling out common factors in the numerator and denominator.
• When multiplying rational expressions, multiply the numerators together and the denominators together.
• When dividing, multiply by the reciprocal of the divisor.

Factors like \((x-y)\) and \(8\) must be identified and canceled where appropriate to simplify the expression further. Through these processes, handling complex rational expressions becomes manageable, ultimately simplifying the process of solving algebraic equations that involve them.

Simplification of Expressions

Simplification in algebra involves reducing expressions to their simplest form. This process can make it easier to work with equations and helps in finding solutions efficiently. Simplification often involves performing operations such as factorization, canceling common terms, and reducing fractions—in other words, making expressions as straightforward as possible.

Within our exercise, simplification was achieved by:

• Recognizing common factors within numerators and denominators and canceling them out.
• Changing division into multiplication by using the reciprocal, which allowed us to multiply and simplify in one step.
• Executing arithmetic operations to combine terms where needed, reducing both the complexity and size of the expression.

The ultimate goal of simplification is to make the expression concise and clear, as seen here where the expression was simplified to \(\frac{xy(xy-1)}{96}\). Through simplification, what once seemed complex becomes more manageable, both in appearance and in solving potential equations.