Question

name
4-3 mathematical literacy and vocabulary
multiplying and dividing rational expressions
match each expression on the left with an equivalent expression on the right.

1. \\(\frac{x}{x + 4} \cdot \frac{x^2 + 8x + 16}{3x}\\) a. \\(\frac{xy}{4}\\)
2. \\(\frac{x^2 - 5x + 6}{2} \div \frac{x - 3}{x - 2}\\) b. \\(\frac{x + 2}{x - 3}\\)
3. \\(\frac{4x^2y^3}{16xy^2}\\) c. \\(\frac{x + 4}{3}\\)
4. \\(\frac{x - 2}{4} \cdot \frac{x + 5}{x - 1}\\) d. \\(\frac{(x - 2)^2}{2}\\)
5. \\(\frac{x^2 + 7x + 10}{x^2 + 2x - 15}\\) e. \\(\frac{x^2 + 3x - 10}{4x - 4}\\)

Explanation:

Step1: Factor quadratic numerator

$\frac{x}{x+4} \cdot \frac{(x+4)^2}{3x}$

Step2: Cancel common factors

$\frac{x \cdot (x+4)^2}{(x+4) \cdot 3x} = \frac{x+4}{3}$
---

Step1: Rewrite division as multiplication

$\frac{x^2 -5x +6}{2} \cdot \frac{x-2}{x-3}$

Step2: Factor quadratic numerator

$\frac{(x-2)(x-3)}{2} \cdot \frac{x-2}{x-3}$

Step3: Cancel common factors

$\frac{(x-2)(x-3)(x-2)}{2(x-3)} = \frac{(x-2)^2}{2}$
---

Step1: Cancel common terms

$\frac{4x^2y^3}{16xy^2} = \frac{4}{16} \cdot \frac{x^2}{x} \cdot \frac{y^3}{y^2} = \frac{xy}{4}$
---

Step1: Multiply numerators and denominators

$\frac{(x-2)(x+5)}{4(x-1)}$

Step2: Expand numerator

$\frac{x^2 +5x -2x -10}{4x -4} = \frac{x^2 +3x -10}{4x -4}$
---

Step1: Factor numerator and denominator

$\frac{(x+2)(x+5)}{(x+5)(x-3)}$

Step2: Cancel common factors

$\frac{(x+2)(x+5)}{(x+5)(x-3)} = \frac{x+2}{x-3}$

Answer:

1. C. $\frac{x+4}{3}$
2. D. $\frac{(x-2)^2}{2}$
3. A. $\frac{xy}{4}$
4. E. $\frac{x^2 + 3x - 10}{4x - 4}$
5. B. $\frac{x+2}{x-3}$