Simplifying Rational Expressions (A) Worksheet | Algebra II PDF ... - Free Printable

To solve the problem of simplifying the given rational expressions, we will simplify each expression step by step. Let's start with Section A and then move on to Section B.

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Section A: Simplify the following algebraic fractions

#### 1. $\frac{x(x + 3)}{x}$
- Factor out $x$ in the numerator.
- Cancel the common factor $x$ (assuming $x \neq 0$).
$$
\frac{x(x + 3)}{x} = x + 3
$$
Answer: $x + 3$

#### 2. $\frac{y(y - 8)}{y}$
- Factor out $y$ in the numerator.
- Cancel the common factor $y$ (assuming $y \neq 0$).
$$
\frac{y(y - 8)}{y} = y - 8
$$
Answer: $y - 8$

#### 3. $\frac{x(x + 5)}{(x + 5)}$
- Cancel the common factor $(x + 5)$ (assuming $x \neq -5$).
$$
\frac{x(x + 5)}{(x + 5)} = x
$$
Answer: $x$

#### 4. $\frac{8(y - 16)}{4}$
- Simplify the constant terms.
$$
\frac{8(y - 16)}{4} = 2(y - 16)
$$
Answer: $2(y - 16)$

#### 5. $\frac{5(x - 7)}{10x(x - 7)}$
- Cancel the common factor $(x - 7)$ (assuming $x \neq 7$).
- Simplify the constants.
$$
\frac{5(x - 7)}{10x(x - 7)} = \frac{5}{10x} = \frac{1}{2x}
$$
Answer: $\frac{1}{2x}$

#### 6. $\frac{3x(3x - 4)}{18x^2(3x - 4)}$
- Cancel the common factors $3x$ and $(3x - 4)$ (assuming $x \neq 0$ and $x \neq \frac{4}{3}$).
- Simplify the constants.
$$
\frac{3x(3x - 4)}{18x^2(3x - 4)} = \frac{1}{6x}
$$
Answer: $\frac{1}{6x}$

#### 7. $\frac{x(x + 5)(x - 5)}{(x + 5)}$
- Cancel the common factor $(x + 5)$ (assuming $x \neq -5$).
$$
\frac{x(x + 5)(x - 5)}{(x + 5)} = x(x - 5)
$$
Answer: $x(x - 5)$

#### 8. $\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)}$
- Cancel the common factors $9y$ and $(y - 1)$ (assuming $y \neq 0$ and $y \neq 1$).
- Simplify the constants.
$$
\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)} = \frac{2y - 1}{3y}
$$
Answer: $\frac{2y - 1}{3y}$

#### 9. $\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)}$
- Cancel the common factors $(x + 1)$ and $(x - 1)$ (assuming $x \neq -1$ and $x \neq 1$).
$$
\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)} = x(x + 1)(x - 1)
$$
Answer: $x(x + 1)(x - 1)$

#### 10. $\frac{8y(y + 4)^2}{12y^2(y + 4)}$
- Cancel the common factors $4y$ and $(y + 4)$ (assuming $y \neq 0$ and $y \neq -4$).
- Simplify the constants.
$$
\frac{8y(y + 4)^2}{12y^2(y + 4)} = \frac{2(y + 4)}{3y}
$$
Answer: $\frac{2(y + 4)}{3y}$

#### 11. $\frac{x(3x - 2)}{7x^3(3x - 2)}$
- Cancel the common factors $x$ and $(3x - 2)$ (assuming $x \neq 0$ and $x \neq \frac{2}{3}$).
- Simplify the constants.
$$
\frac{x(3x - 2)}{7x^3(3x - 2)} = \frac{1}{7x^2}
$$
Answer: $\frac{1}{7x^2}$

#### 12. $\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3}$
- Cancel the common factors $3x^3$ and $(5y - 3)$ (assuming $x \neq 0$ and $y \neq \frac{3}{5}$).
- Simplify the constants.
$$
\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3} = \frac{y + 3}{6x(5y - 3)^2}
$$
Answer: $\frac{y + 3}{6x(5y - 3)^2}$

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Section B: Simplify the following algebraic fractions

#### 1. $\frac{8x + 4}{2}$
- Factor out the common factor in the numerator.
- Simplify.
$$
\frac{8x + 4}{2} = \frac{2(4x + 2)}{2} = 4x + 2
$$
Answer: $4x + 2$

#### 2. $\frac{2y + 6}{4}$
- Factor out the common factor in the numerator.
- Simplify.
$$
\frac{2y + 6}{4} = \frac{2(y + 3)}{4} = \frac{y + 3}{2}
$$
Answer: $\frac{y + 3}{2}$

#### 3. $\frac{7x}{14x - 21}$
- Factor the denominator.
- Simplify.
$$
\frac{7x}{14x - 21} = \frac{7x}{7(2x - 3)} = \frac{x}{2x - 3}
$$
Answer: $\frac{x}{2x - 3}$

#### 4. $\frac{9y^2}{3y + 27y^2}$
- Factor the denominator.
- Simplify.
$$
\frac{9y^2}{3y + 27y^2} = \frac{9y^2}{3y(1 + 9y)} = \frac{3y}{1 + 9y}
$$
Answer: $\frac{3y}{1 + 9y}$

#### 5. $\frac{x - 4}{5x - 20}$
- Factor the denominator.
- Simplify.
$$
\frac{x - 4}{5x - 20} = \frac{x - 4}{5(x - 4)} = \frac{1}{5}
$$
Answer: $\frac{1}{5}$

#### 6. $\frac{6y - 30y^2}{24y^2}$
- Factor the numerator.
- Simplify.
$$
\frac{6y - 30y^2}{24y^2} = \frac{6y(1 - 5y)}{24y^2} = \frac{1 - 5y}{4y}
$$
Answer: $\frac{1 - 5y}{4y}$

#### 7. $\frac{x - 4}{4 - x}$
- Rewrite the denominator as $-(x - 4)$.
- Simplify.
$$
\frac{x - 4}{4 - x} = \frac{x - 4}{-(x - 4)} = -1
$$
Answer: $-1$

#### 8. $\frac{21 - 3x}{42x - 6x^2}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{21 - 3x}{42x - 6x^2} = \frac{3(7 - x)}{6x(7 - x)} = \frac{1}{2x}
$$
Answer: $\frac{1}{2x}$

#### 9. $\frac{x + 2}{x^2 + 7x + 10}$
- Factor the denominator.
- Simplify.
$$
\frac{x + 2}{x^2 + 7x + 10} = \frac{x + 2}{(x + 2)(x + 5)} = \frac{1}{x + 5}
$$
Answer: $\frac{1}{x + 5}$

#### 10. $\frac{x^2 - 13x + 36}{x - 4}$
- Factor the numerator.
- Simplify.
$$
\frac{x^2 - 13x + 36}{x - 4} = \frac{(x - 4)(x - 9)}{x - 4} = x - 9
$$
Answer: $x - 9$

#### 11. $\frac{x^2 - 8x - 20}{9x + 18}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{x^2 - 8x - 20}{9x + 18} = \frac{(x - 10)(x + 2)}{9(x + 2)} = \frac{x - 10}{9}
$$
Answer: $\frac{x - 10}{9}$

#### 12. $\frac{5x + 40}{x^2 + 6x - 16}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{5x + 40}{x^2 + 6x - 16} = \frac{5(x + 8)}{(x + 8)(x - 2)} = \frac{5}{x - 2}
$$
Answer: $\frac{5}{x - 2}$

#### 13. $\frac{12x + 20}{9x^2 + 9x - 10}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{12x + 20}{9x^2 + 9x - 10} = \frac{4(3x + 5)}{(3x + 5)(3x - 2)} = \frac{4}{3x - 2}
$$
Answer: $\frac{4}{3x - 2}$

#### 14. $\frac{x^2 + 5x + 6}{x^2 + 14x + 24}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{x^2 + 5x + 6}{x^2 + 14x + 24} = \frac{(x + 2)(x + 3)}{(x + 2)(x + 12)} = \frac{x + 3}{x + 12}
$$
Answer: $\frac{x + 3}{x + 12}$

#### 15. $\frac{x^2 - 7x - 44}{x^2 - 17x + 66}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{x^2 - 7x - 44}{x^2 - 17x + 66} = \frac{(x - 11)(x + 4)}{(x - 11)(x - 6)} = \frac{x + 4}{x - 6}
$$
Answer: $\frac{x + 4}{x - 6}$

#### 16. $\frac{6x^2 - x - 1}{15x^2 + 8x + 1}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{6x^2 - x - 1}{15x^2 + 8x + 1} = \frac{(3x + 1)(2x - 1)}{(3x + 1)(5x + 1)} = \frac{2x - 1}{5x + 1}
$$
Answer: $\frac{2x - 1}{5x + 1}$

#### 17. $\frac{x^2 - y^2}{(x + y)^2}$
- Factor the numerator using the difference of squares.
- Simplify.
$$
\frac{x^2 - y^2}{(x + y)^2} = \frac{(x - y)(x + y)}{(x + y)^2} = \frac{x - y}{x + y}
$$
Answer: $\frac{x - y}{x + y}$

#### 18. $\frac{4y^2 - 9x^2}{4x^2y + 6x^3}$
- Factor the numerator and denominator.
- Simplify.
$$
\frac{4y^2 - 9x^2}{4x^2y + 6x^3} = \frac{(2y - 3x)(2y + 3x)}{2x^2(2y + 3x)} = \frac{2y - 3x}{2x^2}
$$
Answer: $\frac{2y - 3x}{2x^2}$

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Final Answers:

- Section A:
1. $x + 3$
2. $y - 8$
3. $x$
4. $2(y - 16)$
5. $\frac{1}{2x}$
6. $\frac{1}{6x}$
7. $x(x - 5)$
8. $\frac{2y - 1}{3y}$
9. $x(x + 1)(x - 1)$
10. $\frac{2(y + 4)}{3y}$
11. $\frac{1}{7x^2}$
12. $\frac{y + 3}{6x(5y - 3)^2}$

- Section B:
1. $4x + 2$
2. $\frac{y + 3}{2}$
3. $\frac{x}{2x - 3}$
4. $\frac{3y}{1 + 9y}$
5. $\frac{1}{5}$
6. $\frac{1 - 5y}{4y}$
7. $-1$
8. $\frac{1}{2x}$
9. $\frac{1}{x + 5}$
10. $x - 9$
11. $\frac{x - 10}{9}$
12. $\frac{5}{x - 2}$
13. $\frac{4}{3x - 2}$
14. $\frac{x + 3}{x + 12}$
15. $\frac{x + 4}{x - 6}$
16. $\frac{2x - 1}{5x + 1}$
17. $\frac{x - y}{x + y}$
18. $\frac{2y - 3x}{2x^2}$

Boxed Final Answer:
$$
\boxed{
\text{See detailed solutions above.}
}
$$