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 $):
$$
= x + 3
$$

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2)
$$
\frac{y(y - 8)}{y}
$$
Cancel $ y $ (assuming $ y \neq 0 $):
$$
= y - 8
$$

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3)
$$
\frac{x(x + 5)}{x + 5}
$$
Cancel $ x + 5 $ (assuming $ x \neq -5 $):
$$
= x
$$

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4)
$$
\frac{8(y - 16)}{4}
$$
Simplify constants: $ \frac{8}{4} = 2 $:
$$
= 2(y - 16) = 2y - 32
$$

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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} $:
$$
= \frac{1}{2x}
$$

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6)
$$
\frac{3x(3x - 4)}{18x^2(3x - 4)}
$$
Cancel $ 3x $ and $ 3x - 4 $ (assuming $ x \neq 0 $, $ 3x - 4 \neq 0 $):
$$
= \frac{1}{6x}
$$

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7)
$$
\frac{x(x + 5)(x - 5)}{x + 5}
$$
Cancel $ x + 5 $ (assuming $ x \neq -5 $):
$$
= x(x - 5)
$$

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8)
$$
\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)}
$$
Cancel $ y $, $ y - 1 $, and simplify $ \frac{9}{27} = \frac{1}{3} $:
$$
= \frac{(2y - 1)}{3y}
$$

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9)
$$
\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)}
$$
Cancel one $ (x+1) $ and $ (x-1) $:
$$
= x(x + 1)(x - 1)
$$

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10)
$$
\frac{8y(y + 4)^2}{12y^2(y + 4)}
$$
Cancel $ y $, $ y + 4 $, and simplify $ \frac{8}{12} = \frac{2}{3} $:
$$
= \frac{2(y + 4)}{3y}
$$

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11)
$$
\frac{x(3x - 2)}{7x^3(3x - 2)^2}
$$
Cancel $ x $, $ 3x - 2 $:
$$
= \frac{1}{7x^2(3x - 2)}
$$

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12)
$$
\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3}
$$
Cancel $ 3x^3 $, $ 5y - 3 $:
$$
= \frac{(y + 3)}{6x(5y - 3)^2}
$$

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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 by 2:
$$
= 4x + 2
$$

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2)
$$
\frac{2y + 6}{4}
$$
Factor: $ \frac{2(y + 3)}{4} = \frac{y + 3}{2} $

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3)
$$
\frac{7x}{14x - 21}
$$
Factor denominator: $ 7(2x - 3) $, cancel 7:
$$
= \frac{x}{2x - 3}
$$

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4)
$$
\frac{9y^2}{3y + 27y^2}
$$
Rewrite denominator: $ 3y(1 + 9y) $, numerator: $ 9y^2 $
$$
= \frac{9y^2}{3y(1 + 9y)} = \frac{3y}{1 + 9y}
$$

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5)
$$
\frac{x - 4}{5x - 20}
$$
Factor denominator: $ 5(x - 4) $:
$$
= \frac{x - 4}{5(x - 4)} = \frac{1}{5}, \quad x \ne 4
$$

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6)
$$
\frac{6y - 30y^2}{24y^2}
$$
Factor numerator: $ 6y(1 - 5y) $, denominator: $ 24y^2 $:
$$
= \frac{6y(1 - 5y)}{24y^2} = \frac{1 - 5y}{4y}
$$

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7)
$$
\frac{x - 4}{4 - x}
$$
Note: $ 4 - x = -(x - 4) $:
$$
= \frac{x - 4}{-(x - 4)} = -1, \quad x \ne 4
$$

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8)
$$
\frac{21 - 3x}{42x - 6x^2}
$$
Numerator: $ 3(7 - x) $, denominator: $ 6x(7 - x) $:
$$
= \frac{3(7 - x)}{6x(7 - x)} = \frac{3}{6x} = \frac{1}{2x}, \quad x \ne 7
$$

---

9)
$$
\frac{x + 2}{x^2 + 7x + 10}
$$
Factor denominator: $ (x + 2)(x + 5) $:
$$
= \frac{x + 2}{(x + 2)(x + 5)} = \frac{1}{x + 5}, \quad x \ne -2
$$

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10)
$$
\frac{x^2 - 13x + 36}{x - 4}
$$
Factor numerator: $ x^2 - 13x + 36 = (x - 4)(x - 9) $:
$$
= \frac{(x - 4)(x - 9)}{x - 4} = x - 9, \quad x \ne 4
$$

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11)
$$
\frac{x^2 - 8x - 20}{9x + 18}
$$
Factor numerator: $ (x - 10)(x + 2) $, denominator: $ 9(x + 2) $:
$$
= \frac{(x - 10)(x + 2)}{9(x + 2)} = \frac{x - 10}{9}, \quad x \ne -2
$$

---

12)
$$
\frac{5x + 40}{x^2 + 6x - 16}
$$
Numerator: $ 5(x + 8) $, denominator: $ (x + 8)(x - 2) $:
$$
= \frac{5(x + 8)}{(x + 8)(x - 2)} = \frac{5}{x - 2}, \quad x \ne -8
$$

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13)
$$
\frac{12x + 20}{9x^2 + 9x - 10}
$$
Numerator: $ 4(3x + 5) $, denominator: factor $ 9x^2 + 9x - 10 $.
Try: $ (3x - 2)(3x + 5) = 9x^2 + 15x - 6x - 10 = 9x^2 + 9x - 10 $ ✔
So:
$$
= \frac{4(3x + 5)}{(3x - 2)(3x + 5)} = \frac{4}{3x - 2}, \quad x \ne -\frac{5}{3}
$$

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14)
$$
\frac{x^2 + 5x + 6}{x^2 + 14x + 24}
$$
Factor:
Numerator: $ (x + 2)(x + 3) $,
Denominator: $ (x + 2)(x + 12) $:
$$
= \frac{(x + 2)(x + 3)}{(x + 2)(x + 12)} = \frac{x + 3}{x + 12}, \quad x \ne -2
$$

---

15)
$$
\frac{x^2 - 7x - 44}{x^2 - 17x + 66}
$$
Factor:
Numerator: $ x^2 - 7x - 44 = (x - 11)(x + 4) $,
Denominator: $ x^2 - 17x + 66 = (x - 11)(x - 6) $:
$$
= \frac{(x - 11)(x + 4)}{(x - 11)(x - 6)} = \frac{x + 4}{x - 6}, \quad x \ne 11
$$

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16)
$$
\frac{6x^2 - x - 1}{15x^2 + 8x + 1}
$$
Factor both:
Numerator: $ 6x^2 - x - 1 = (3x + 1)(2x - 1) $
Denominator: $ 15x^2 + 8x + 1 = (3x + 1)(5x + 1) $
$$
= \frac{(3x + 1)(2x - 1)}{(3x + 1)(5x + 1)} = \frac{2x - 1}{5x + 1}, \quad x \ne -\frac{1}{3}
$$

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17)
$$
\frac{x^2 - y^2}{(x + y)^2}
$$
Numerator: $ (x - y)(x + y) $, denominator: $ (x + y)^2 $:
$$
= \frac{(x - y)(x + y)}{(x + y)^2} = \frac{x - y}{x + y}, \quad x \ne -y
$$

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18)
$$
\frac{4y^2 - 9x^2}{4x^2y + 6x^3}
$$
Numerator: difference of squares: $ (2y - 3x)(2y + 3x) $
Denominator: factor $ 2x^2(2y + 3x) $:
$$
= \frac{(2y - 3x)(2y + 3x)}{2x^2(2y + 3x)} = \frac{2y - 3x}{2x^2}, \quad 2y + 3x \ne 0
$$

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✔ Final Answers:

#### Section A
1. $ x + 3 $
2. $ y - 8 $
3. $ x $
4. $ 2y - 32 $
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} $

> ⚠️ Remember: All answers are valid only when the original expression is defined (i.e., denominators ≠ 0). Exclude values that make any denominator zero.

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