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

Let's solve and simplify each of the rational expressions step by step. We'll go through Section A and Section B, simplifying each expression by factoring, canceling common terms, and reducing fractions.

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✔ Section A: Simplify the following algebraic fractions

We simplify by canceling common factors in numerator and denominator.

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#### 1) $\frac{x(x + 3)}{x}$

- Cancel $x$ (assuming $x \neq 0$):
$$
= x + 3
$$

✔ Answer: $x + 3$

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#### 2) $\frac{y(y - 8)}{y}$

- Cancel $y$ (assuming $y \neq 0$):
$$
= y - 8
$$

✔ Answer: $y - 8$

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#### 3) $\frac{x(x + 5)}{(x + 5)}$

- Cancel $(x + 5)$ (assuming $x \neq -5$):
$$
= x
$$

✔ Answer: $x$

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#### 4) $\frac{8(y - 16)}{4}$

- Simplify constants: $\frac{8}{4} = 2$
$$
= 2(y - 16)
$$

✔ Answer: $2(y - 16)$

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#### 5) $\frac{5(x - 7)}{10x(x - 7)}$

- Cancel $(x - 7)$ (assuming $x \neq 7$), and simplify $\frac{5}{10x} = \frac{1}{2x}$:
$$
= \frac{1}{2x}
$$

✔ Answer: $\frac{1}{2x}$

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#### 6) $\frac{3x(3x - 4)}{18x^2(3x - 4)}$

- Cancel $(3x - 4)$ (assuming $3x \neq 4$), and simplify $\frac{3x}{18x^2} = \frac{1}{6x}$:
$$
= \frac{1}{6x}
$$

✔ Answer: $\frac{1}{6x}$

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#### 7) $\frac{x(x + 5)(x - 5)}{(x + 5)}$

- Cancel $(x + 5)$ (assuming $x \neq -5$):
$$
= x(x - 5)
$$

✔ Answer: $x(x - 5)$

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#### 8) $\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)}$

- Cancel $(y - 1)$ (assuming $y \neq 1$), and simplify:
- $\frac{9y}{27y^2} = \frac{1}{3y}$
- So: $\frac{1}{3y}(2y - 1)$
$$
= \frac{2y - 1}{3y}
$$

✔ Answer: $\frac{2y - 1}{3y}$

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#### 9) $\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)}$

- Cancel one $(x + 1)$ and one $(x - 1)$ from numerator and denominator:
$$
= x(x + 1)(x - 1)
$$

✔ Answer: $x(x + 1)(x - 1)$

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#### 10) $\frac{8y(y + 4)^2}{12y^2(y + 4)}$

- Cancel one $(y + 4)$, and simplify:
- $\frac{8y}{12y^2} = \frac{2}{3y}$
- Remaining: $(y + 4)$
$$
= \frac{2(y + 4)}{3y}
$$

✔ Answer: $\frac{2(y + 4)}{3y}$

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#### 11) $\frac{x(3x - 2)}{7x^3(3x - 2)^2}$

- Cancel $(3x - 2)$, and simplify:
- $\frac{x}{7x^3} = \frac{1}{7x^2}$
- Remaining: $\frac{1}{(3x - 2)}$
$$
= \frac{1}{7x^2(3x - 2)}
$$

✔ Answer: $\frac{1}{7x^2(3x - 2)}$

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#### 12) $\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3}$

- Cancel $(5y - 3)$, and simplify:
- $\frac{3x^3}{18x^4} = \frac{1}{6x}$
- Remaining: $\frac{y + 3}{(5y - 3)^2}$
$$
= \frac{y + 3}{6x(5y - 3)^2}
$$

✔ Answer: $\frac{y + 3}{6x(5y - 3)^2}$

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✔ Section B: Simplify the following algebraic fractions

Now we factor and simplify.

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#### 1) $\frac{8x + 4}{2}$

- Factor numerator: $4(2x + 1)$
- Divide: $\frac{4(2x + 1)}{2} = 2(2x + 1)$
$$
= 4x + 2
$$

✔ Answer: $4x + 2$

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#### 2) $\frac{2y + 6}{4}$

- Factor numerator: $2(y + 3)$
- Divide: $\frac{2(y + 3)}{4} = \frac{y + 3}{2}$

✔ Answer: $\frac{y + 3}{2}$

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#### 3) $\frac{7x}{14x - 21}$

- Factor denominator: $7(2x - 3)$
- Numerator: $7x$
- Cancel 7:
$$
= \frac{x}{2x - 3}
$$

✔ Answer: $\frac{x}{2x - 3}$

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#### 4) $\frac{9y^2}{3y + 27y^2}$

- Denominator: $3y + 27y^2 = 3y(1 + 9y)$
- Numerator: $9y^2 = 9y^2$
- Write:
$$
= \frac{9y^2}{3y(1 + 9y)} = \frac{3y}{1 + 9y}
$$

✔ Answer: $\frac{3y}{1 + 9y}$

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#### 5) $\frac{x - 4}{5x - 20}$

- Denominator: $5(x - 4)$
- Cancel $(x - 4)$ (assume $x \neq 4$):
$$
= \frac{1}{5}
$$

✔ Answer: $\frac{1}{5}$

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#### 6) $\frac{6y - 30y^2}{24y^2}$

- Factor numerator: $6y(1 - 5y)$
- Denominator: $24y^2$
- Simplify:
$$
= \frac{6y(1 - 5y)}{24y^2} = \frac{1 - 5y}{4y}
$$

✔ Answer: $\frac{1 - 5y}{4y}$

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#### 7) $\frac{x - 4}{4 - x}$

- Note: $4 - x = -(x - 4)$
- So:
$$
= \frac{x - 4}{-(x - 4)} = -1 \quad \text{(for } x \neq 4\text{)}
$$

✔ Answer: $-1$

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#### 8) $\frac{21 - 3x}{42x - 6x^2}$

- Numerator: $-3(x - 7)$ or $3(7 - x)$
- Denominator: $6x(7 - x)$
- Write numerator: $3(7 - x)$
- Denominator: $6x(7 - x)$
- Cancel $(7 - x)$:
$$
= \frac{3}{6x} = \frac{1}{2x}
$$

✔ Answer: $\frac{1}{2x}$

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#### 9) $\frac{x + 2}{x^2 + 7x + 10}$

- Factor denominator: $x^2 + 7x + 10 = (x + 2)(x + 5)$
- Cancel $(x + 2)$:
$$
= \frac{1}{x + 5}
$$

✔ Answer: $\frac{1}{x + 5}$

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#### 10) $\frac{x^2 - 13x + 36}{x - 4}$

- Factor numerator: $x^2 - 13x + 36 = (x - 4)(x - 9)$
- Cancel $(x - 4)$:
$$
= x - 9
$$

✔ Answer: $x - 9$

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#### 11) $\frac{x^2 - 8x - 20}{9x + 18}$

- Factor numerator: $x^2 - 8x - 20 = (x - 10)(x + 2)$
- Denominator: $9(x + 2)$
- Cancel $(x + 2)$:
$$
= \frac{x - 10}{9}
$$

✔ Answer: $\frac{x - 10}{9}$

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#### 12) $\frac{5x + 40}{x^2 + 6x - 16}$

- Numerator: $5(x + 8)$
- Denominator: $x^2 + 6x - 16 = (x + 8)(x - 2)$
- Cancel $(x + 8)$:
$$
= \frac{5}{x - 2}
$$

✔ Answer: $\frac{5}{x - 2}$

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#### 13) $\frac{12x + 20}{9x^2 + 9x - 10}$

- Numerator: $4(3x + 5)$
- Denominator: Try factoring:
- $9x^2 + 9x - 10$: Find two numbers that multiply to $-90$, add to $9$: $15$ and $-6$
- $9x^2 + 15x - 6x - 10 = 3x(3x + 5) - 2(3x + 5) = (3x - 2)(3x + 5)$
- So:
$$
= \frac{4(3x + 5)}{(3x - 2)(3x + 5)} = \frac{4}{3x - 2}
$$

✔ Answer: $\frac{4}{3x - 2}$

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#### 14) $\frac{x^2 + 5x + 6}{x^2 + 14x + 24}$

- Numerator: $(x + 2)(x + 3)$
- Denominator: $(x + 2)(x + 12)$
- Cancel $(x + 2)$:
$$
= \frac{x + 3}{x + 12}
$$

✔ Answer: $\frac{x + 3}{x + 12}$

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#### 15) $\frac{x^2 - 7x - 44}{x^2 - 17x + 66}$

- Numerator: $x^2 - 7x - 44 = (x - 11)(x + 4)$
- Denominator: $x^2 - 17x + 66 = (x - 11)(x - 6)$
- Cancel $(x - 11)$:
$$
= \frac{x + 4}{x - 6}
$$

✔ Answer: $\frac{x + 4}{x - 6}$

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#### 16) $\frac{6x^2 - x - 1}{15x^2 + 8x + 1}$

- Factor numerator: $6x^2 - x - 1 = (3x + 1)(2x - 1)$
- Denominator: $15x^2 + 8x + 1 = (5x + 1)(3x + 1)$
- Cancel $(3x + 1)$:
$$
= \frac{2x - 1}{5x + 1}
$$

✔ Answer: $\frac{2x - 1}{5x + 1}$

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#### 17) $\frac{x^2 - y^2}{(x + y)^2}$

- Numerator: difference of squares: $(x - y)(x + y)$
- Denominator: $(x + y)^2$
- Cancel one $(x + y)$:
$$
= \frac{x - y}{x + y}
$$

✔ Answer: $\frac{x - y}{x + y}$

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#### 18) $\frac{4y^2 - 9x^2}{4x^2y + 6x^3}$

- Numerator: $4y^2 - 9x^2 = (2y - 3x)(2y + 3x)$
- Denominator: $2x^2(2y + 3x)$
- Cancel $(2y + 3x)$:
$$
= \frac{2y - 3x}{2x^2}
$$

✔ Answer: $\frac{2y - 3x}{2x^2}$

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✔ Final Answers Summary:

#### 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. $\frac{1}{7x^2(3x - 2)}$
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. $\frac{2y - 3x}{2x^2}$

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