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Domain and Simplification of Rational Expression $\\left( \\frac{x}{xy - y^2} - \\frac{y}{x^2 - xy} \\right) \\div \\frac{x^2 - y^2}{8xy}$
Mathematics (Algebra)
Grade 10 (High School)
Question Content
Find the domain and simplify the expression: $\\left( \\frac{x}{xy - y^2} - \\frac{y}{x^2 - xy} \\right) \\div \\frac{x^2 - y^2}{8xy}$
Correct Answer
Domain: $x \\neq 0, y \\neq 0, x \\neq y, x \\neq -y$; Simplified expression: $\\frac{8}{x - y}$
Detailed Solution Steps
1
Step 1: Determine the domain by identifying all values that make any denominator zero or the divisor zero: \n- $xy - y^2 = y(x - y) \\neq 0 \\implies y \\neq 0, x \\neq y$\n- $x^2 - xy = x(x - y) \\neq 0 \\implies x \\neq 0, x \\neq y$\n- Divisor $\\frac{x^2 - y^2}{8xy} \\neq 0 \\implies (x-y)(x+y) \\neq 0$ and $8xy \\neq 0 \\implies x \\neq \\pm y, x \\neq 0, y \\neq 0$\nCombining these, the domain is all real numbers $x,y$ where $x \\neq 0, y \\neq 0, x \\neq y, x \\neq -y$.
2
Step 2: Factor all polynomial terms in the expression:\n- $xy - y^2 = y(x - y)$\n- $x^2 - xy = x(x - y)$\n- $x^2 - y^2 = (x - y)(x + y)$\nRewrite the expression with factored forms: $\\left( \\frac{x}{y(x - y)} - \\frac{y}{x(x - y)} \\right) \\div \\frac{(x - y)(x + y)}{8xy}$
3
Step 3: Combine the two fractions inside the parentheses using a common denominator $xy(x - y)$:\n$\\frac{x \\cdot x - y \\cdot y}{xy(x - y)} = \\frac{x^2 - y^2}{xy(x - y)}$. Substitute $x^2 - y^2 = (x - y)(x + y)$, then cancel $(x - y)$ (allowed since $x \\neq y$): $\\frac{x + y}{xy}$
4
Step 4: Convert division to multiplication by the reciprocal of the divisor:\n$\\frac{x + y}{xy} \\times \\frac{8xy}{(x - y)(x + y)}$
5
Step 5: Cancel common factors $(x + y)$ and $xy$ from numerator and denominator, leaving the simplified expression: $\\frac{8}{x - y}$
Knowledge Points Involved
1
Domain of Rational Expressions
The set of all valid input values (variables) for which a rational expression is defined. A rational expression is undefined when any denominator equals zero, or when a divisor (fraction) equals zero (its numerator is zero). For this problem, we exclude values that make any denominator zero or the divisor zero.
2
Factoring Polynomials
Breaking down polynomials into simpler factors. Key techniques used here: factoring out the greatest common factor (GCF) (e.g., $xy - y^2 = y(x - y)$) and factoring the difference of squares ($a^2 - b^2 = (a - b)(a + b)$) for $x^2 - y^2$.
3
Combining Rational Expressions
To add or subtract rational expressions, find a common denominator (the least common multiple of the denominators), rewrite each fraction with the common denominator, then combine the numerators. Here, we used $xy(x - y)$ as the common denominator for the two fractions inside the parentheses.
4
Dividing Rational Expressions
Dividing by a rational expression is equivalent to multiplying by its reciprocal. For $A \\div \\frac{B}{C}$, this becomes $A \\times \\frac{C}{B}$, provided $B \\neq 0$ and $C \\neq 0$. We applied this to convert the division operation into multiplication, simplifying the problem further.
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