- Start: $\frac{x - 2}{x + 5} \cdot \frac{x + 5}{x - 1} = \frac{x - 2}{x - 1}$
- Next: $\frac{x - 2}{x - 1} \cdot (x - 1) = x - 2$ (This step uses the expression $(x-2)(x-1)$ as a multiplier, which is incorrect. The correct path from $\frac{x-2}{x-1}$ leads to the division problem $\frac{x+6}{x-6} \div \frac{x^2 + 4x - 12}{x - 6}$.)
- Correction: From $\frac{x - 2}{x - 1}$, follow the path to $\frac{x + 6}{x - 6} \div \frac{x^2 + 4x - 12}{x - 6}$
- Simplify the divisor: $x^2 + 4x - 12 = (x + 6)(x - 2)$
- Rewrite division as multiplication: $\frac{x + 6}{x - 6} \cdot \frac{x - 6}{(x + 6)(x - 2)}$
- Cancel common factors: $\frac{1}{x - 2}$
- Next: $\frac{1}{x - 2} \cdot \frac{x + 6}{x - 2} = \frac{x + 6}{(x - 2)^2}$ (This step uses the expression $\frac{x+6}{x-2}$ as a multiplier, which is incorrect. The correct path from $\frac{1}{x-2}$ leads to the problem $\frac{2x - 10}{x^2 - 4x - 5} \div \frac{6x - 18}{x + 1}$.)
- Correction: From $\frac{1}{x - 2}$, follow the path to $\frac{2x - 10}{x^2 - 4x - 5} \div \frac{6x - 18}{x + 1}$
- Factor all expressions:
- Numerator of first fraction: $2x - 10 = 2(x - 5)$
- Denominator of first fraction: $x^2 - 4x - 5 = (x - 5)(x + 1)$
- Numerator of second fraction: $6x - 18 = 6(x - 3)$
- Rewrite division as multiplication: $\frac{2(x - 5)}{(x - 5)(x + 1)} \cdot \frac{x + 1}{6(x - 3)}$
- Cancel common factors: $\frac{2}{6(x - 3)} = \frac{1}{3(x - 3)}$
- Next: $\frac{1}{3(x - 3)} \cdot (x + 1) = \frac{x + 1}{3(x - 3)}$ (This step uses the expression $x+1$ as a multiplier, which is incorrect. The correct path from $\frac{1}{3(x-3)}$ leads to the problem $\frac{x^2 + 2x - 3}{x^2 + 4x + 3} \div (x - 1)$.)
- Correction: From $\frac{1}{3(x - 3)}$, follow the path to $\frac{x^2 + 2x - 3}{x^2 + 4x + 3} \div (x - 1)$
- Factor all expressions:
- Numerator: $x^2 + 2x - 3 = (x + 3)(x - 1)$
- Denominator: $x^2 + 4x + 3 = (x + 3)(x + 1)$
- Rewrite division as multiplication: $\frac{(x + 3)(x - 1)}{(x + 3)(x + 1)} \cdot \frac{1}{x - 1}$
- Cancel common factors: $\frac{1}{x + 1}$
- Next: $\frac{1}{x + 1} \cdot \frac{1}{x + 1} = \frac{1}{(x + 1)^2}$ (This step uses the expression $\frac{1}{x+1}$ as a multiplier, which is incorrect. The correct path from $\frac{1}{x+1}$ leads to the problem $\frac{x^2 - 4}{x + 3} \cdot \frac{x + 2}{x + 3}$.)
- Correction: From $\frac{1}{x + 1}$, follow the path to $\frac{x^2 - 4}{x + 3} \cdot \frac{x + 2}{x + 3}$
- Factor the numerator: $x^2 - 4 = (x + 2)(x - 2)$
- Multiply the fractions: $\frac{(x + 2)(x - 2)}{x + 3} \cdot \frac{x + 2}{x + 3} = \frac{(x + 2)^2(x - 2)}{(x + 3)^2}$
- Final Answer: $\frac{(x + 2)^2(x - 2)}{(x + 3)^2}$
Parent Tip: Review the logic above to help your child master the concept of multiplying rational expression worksheet.
