Rational Expression Worksheet #5: Multiply and Divide Rational Expressions with Factoring

Problem: Solve the given rational expressions by multiplying or dividing, and show all work. Factor where necessary.

#### 1. $\frac{2a^3b}{b^3c} \cdot \frac{b}{a}$

Solution:
1. Combine the fractions:
$$
\frac{2a^3b}{b^3c} \cdot \frac{b}{a} = \frac{2a^3b \cdot b}{b^3c \cdot a}
$$
2. Simplify the numerator and denominator:
$$
\frac{2a^3b^2}{ab^3c}
$$
3. Cancel common factors:
- $a$ in the numerator and denominator cancels out.
- $b^2$ in the numerator and $b^3$ in the denominator leaves $b$ in the denominator.
$$
\frac{2a^2}{bc}
$$

Final Answer:
$$
\boxed{\frac{2a^2}{bc}}
$$

---

#### 2. $\frac{y^2 - 2y - 15}{4} \cdot \frac{8}{y + 3}$

Solution:
1. Factor the quadratic expression in the numerator of the first fraction:
$$
y^2 - 2y - 15 = (y - 5)(y + 3)
$$
So, the expression becomes:
$$
\frac{(y - 5)(y + 3)}{4} \cdot \frac{8}{y + 3}
$$
2. Combine the fractions:
$$
\frac{(y - 5)(y + 3) \cdot 8}{4(y + 3)}
$$
3. Cancel common factors:
- $(y + 3)$ in the numerator and denominator cancels out.
- Simplify $8/4$ to $2$.
$$
2(y - 5)
$$

Final Answer:
$$
\boxed{2(y - 5)}
$$

---

#### 3. $\frac{x - 5}{6} \div \frac{2x - 10}{12}$

Solution:
1. Rewrite division as multiplication by the reciprocal:
$$
\frac{x - 5}{6} \div \frac{2x - 10}{12} = \frac{x - 5}{6} \cdot \frac{12}{2x - 10}
$$
2. Factor the denominator of the second fraction:
$$
2x - 10 = 2(x - 5)
$$
So, the expression becomes:
$$
\frac{x - 5}{6} \cdot \frac{12}{2(x - 5)}
$$
3. Combine the fractions:
$$
\frac{(x - 5) \cdot 12}{6 \cdot 2(x - 5)}
$$
4. Cancel common factors:
- $(x - 5)$ in the numerator and denominator cancels out.
- Simplify $12/(6 \cdot 2)$ to $1$.
$$
1
$$

Final Answer:
$$
\boxed{1}
$$

---

#### 4. $\frac{5n + 15}{4n + 8} \cdot \frac{2n + 4}{3n + 9}$

Solution:
1. Factor each term:
- $5n + 15 = 5(n + 3)$
- $4n + 8 = 4(n + 2)$
- $2n + 4 = 2(n + 2)$
- $3n + 9 = 3(n + 3)$
So, the expression becomes:
$$
\frac{5(n + 3)}{4(n + 2)} \cdot \frac{2(n + 2)}{3(n + 3)}
$$
2. Combine the fractions:
$$
\frac{5(n + 3) \cdot 2(n + 2)}{4(n + 2) \cdot 3(n + 3)}
$$
3. Cancel common factors:
- $(n + 3)$ in the numerator and denominator cancels out.
- $(n + 2)$ in the numerator and denominator cancels out.
$$
\frac{5 \cdot 2}{4 \cdot 3} = \frac{10}{12} = \frac{5}{6}
$$

Final Answer:
$$
\boxed{\frac{5}{6}}
$$

---

#### 5. $\frac{x^2 - 2x}{6} \div \frac{3x - 6}{x}$

Solution:
1. Rewrite division as multiplication by the reciprocal:
$$
\frac{x^2 - 2x}{6} \div \frac{3x - 6}{x} = \frac{x^2 - 2x}{6} \cdot \frac{x}{3x - 6}
$$
2. Factor each term:
- $x^2 - 2x = x(x - 2)$
- $3x - 6 = 3(x - 2)$
So, the expression becomes:
$$
\frac{x(x - 2)}{6} \cdot \frac{x}{3(x - 2)}
$$
3. Combine the fractions:
$$
\frac{x(x - 2) \cdot x}{6 \cdot 3(x - 2)}
$$
4. Cancel common factors:
- $(x - 2)$ in the numerator and denominator cancels out.
$$
\frac{x^2}{18}
$$

Final Answer:
$$
\boxed{\frac{x^2}{18}}
$$

---

#### 6. $\frac{m^2 - 2m - 8}{8m + 24} \div \frac{2m - 8}{m^2 + 7m + 12}$

Solution:
1. Rewrite division as multiplication by the reciprocal:
$$
\frac{m^2 - 2m - 8}{8m + 24} \div \frac{2m - 8}{m^2 + 7m + 12} = \frac{m^2 - 2m - 8}{8m + 24} \cdot \frac{m^2 + 7m + 12}{2m - 8}
$$
2. Factor each term:
- $m^2 - 2m - 8 = (m - 4)(m + 2)$
- $8m + 24 = 8(m + 3)$
- $2m - 8 = 2(m - 4)$
- $m^2 + 7m + 12 = (m + 3)(m + 4)$
So, the expression becomes:
$$
\frac{(m - 4)(m + 2)}{8(m + 3)} \cdot \frac{(m + 3)(m + 4)}{2(m - 4)}
$$
3. Combine the fractions:
$$
\frac{(m - 4)(m + 2)(m + 3)(m + 4)}{8(m + 3) \cdot 2(m - 4)}
$$
4. Cancel common factors:
- $(m - 4)$ in the numerator and denominator cancels out.
- $(m + 3)$ in the numerator and denominator cancels out.
$$
\frac{(m + 2)(m + 4)}{16}
$$

Final Answer:
$$
\boxed{\frac{(m + 2)(m + 4)}{16}}
$$

---

#### 7. $\frac{x + 3}{10x + 20} \cdot \frac{x + 2}{x^2 + 4x + 3}$

Solution:
1. Factor each term:
- $10x + 20 = 10(x + 2)$
- $x^2 + 4x + 3 = (x + 3)(x + 1)$
So, the expression becomes:
$$
\frac{x + 3}{10(x + 2)} \cdot \frac{x + 2}{(x + 3)(x + 1)}
$$
2. Combine the fractions:
$$
\frac{(x + 3)(x + 2)}{10(x + 2)(x + 3)(x + 1)}
$$
3. Cancel common factors:
- $(x + 3)$ in the numerator and denominator cancels out.
- $(x + 2)$ in the numerator and denominator cancels out.
$$
\frac{1}{10(x + 1)}
$$

Final Answer:
$$
\boxed{\frac{1}{10(x + 1)}}
$$

---

#### 8. $\frac{x^2 - x - 12}{x - 4} \div \frac{2x + 6}{x - 5}$

Solution:
1. Rewrite division as multiplication by the reciprocal:
$$
\frac{x^2 - x - 12}{x - 4} \div \frac{2x + 6}{x - 5} = \frac{x^2 - x - 12}{x - 4} \cdot \frac{x - 5}{2x + 6}
$$
2. Factor each term:
- $x^2 - x - 12 = (x - 4)(x + 3)$
- $2x + 6 = 2(x + 3)$
So, the expression becomes:
$$
\frac{(x - 4)(x + 3)}{x - 4} \cdot \frac{x - 5}{2(x + 3)}
$$
3. Combine the fractions:
$$
\frac{(x - 4)(x + 3)(x - 5)}{(x - 4) \cdot 2(x + 3)}
$$
4. Cancel common factors:
- $(x - 4)$ in the numerator and denominator cancels out.
- $(x + 3)$ in the numerator and denominator cancels out.
$$
\frac{x - 5}{2}
$$

Final Answer:
$$
\boxed{\frac{x - 5}{2}}
$$

---

#### 9. $\frac{x^2 - 5x - 6}{5x + 15} \div \frac{x^2 - 3x - 4}{7x + 21}$

Solution:
1. Rewrite division as multiplication by the reciprocal:
$$
\frac{x^2 - 5x - 6}{5x + 15} \div \frac{x^2 - 3x - 4}{7x + 21} = \frac{x^2 - 5x - 6}{5x + 15} \cdot \frac{7x + 21}{x^2 - 3x - 4}
$$
2. Factor each term:
- $x^2 - 5x - 6 = (x - 6)(x + 1)$
- $5x + 15 = 5(x + 3)$
- $x^2 - 3x - 4 = (x - 4)(x + 1)$
- $7x + 21 = 7(x + 3)$
So, the expression becomes:
$$
\frac{(x - 6)(x + 1)}{5(x + 3)} \cdot \frac{7(x + 3)}{(x - 4)(x + 1)}
$$
3. Combine the fractions:
$$
\frac{(x - 6)(x + 1) \cdot 7(x + 3)}{5(x + 3) \cdot (x - 4)(x + 1)}
$$
4. Cancel common factors:
- $(x + 1)$ in the numerator and denominator cancels out.
- $(x + 3)$ in the numerator and denominator cancels out.
$$
\frac{7(x - 6)}{5(x - 4)}
$$

Final Answer:
$$
\boxed{\frac{7(x - 6)}{5(x - 4)}}
$$

---

#### 10. $\frac{24x^3}{25y^3} \cdot \frac{15y^2}{8x^2}$

Solution:
1. Combine the fractions:
$$
\frac{24x^3}{25y^3} \cdot \frac{15y^2}{8x^2} = \frac{24x^3 \cdot 15y^2}{25y^3 \cdot 8x^2}
$$
2. Simplify the coefficients and variables:
- Coefficients: $24 \cdot 15 = 360$, $25 \cdot 8 = 200$, so $\frac{360}{200} = \frac{9}{5}$.
- Variables: $x^3 / x^2 = x$, $y^2 / y^3 = 1/y$.
$$
\frac{9x}{5y}
$$

Final Answer:
$$
\boxed{\frac{9x}{5y}}
$$

---

#### 11. $\frac{6x - 18}{4x} \cdot \frac{x}{2x - 6}$

Solution:
1. Factor each term:
- $6x - 18 = 6(x - 3)$
- $2x - 6 = 2(x - 3)$
So, the expression becomes:
$$
\frac{6(x - 3)}{4x} \cdot \frac{x}{2(x - 3)}
$$
2. Combine the fractions:
$$
\frac{6(x - 3) \cdot x}{4x \cdot 2(x - 3)}
$$
3. Cancel common factors:
- $(x - 3)$ in the numerator and denominator cancels out.
- $x$ in the numerator and denominator cancels out.
- Simplify $6/(4 \cdot 2)$ to $3/4$.
$$
\frac{3}{4}
$$

Final Answer:
$$
\boxed{\frac{3}{4}}
$$

---

#### 12. $\frac{3x + 12}{12x} \div \frac{x + 4}{48x^3}$

Solution:
1. Rewrite division as multiplication by the reciprocal:
$$
\frac{3x + 12}{12x} \div \frac{x + 4}{48x^3} = \frac{3x + 12}{12x} \cdot \frac{48x^3}{x + 4}
$$
2. Factor each term:
- $3x + 12 = 3(x + 4)$
So, the expression becomes:
$$
\frac{3(x + 4)}{12x} \cdot \frac{48x^3}{x + 4}
$$
3. Combine the fractions:
$$
\frac{3(x + 4) \cdot 48x^3}{12x \cdot (x + 4)}
$$
4. Cancel common factors:
- $(x + 4)$ in the numerator and denominator cancels out.
- Simplify $3 \cdot 48 / 12$ to $12$.
- Simplify $x^3 / x$ to $x^2$.
$$
12x^2
$$

Final Answer:
$$
\boxed{12x^2}
$$

---

Final Answers:

1. $\boxed{\frac{2a^2}{bc}}$
2. $\boxed{2(y - 5)}$
3. $\boxed{1}$
4. $\boxed{\frac{5}{6}}$
5. $\boxed{\frac{x^2}{18}}$
6. $\boxed{\frac{(m + 2)(m + 4)}{16}}$
7. $\boxed{\frac{1}{10(x + 1)}}$
8. $\boxed{\frac{x - 5}{2}}$
9. $\boxed{\frac{7(x - 6)}{5(x - 4)}}$
10. $\boxed{\frac{9x}{5y}}$
11. $\boxed{\frac{3}{4}}$
12. $\boxed{12x^2}$