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.

---

✔ Section A: Simplify the following algebraic fractions

We simplify by canceling common factors in numerator and denominator.

---

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

Cancel $x$ (assuming $x \neq 0$):

$$
= x + 3
$$

✔ Answer: $x + 3$

---

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

Cancel $y$ (assuming $y \neq 0$):

$$
= y - 8
$$

✔ Answer: $y - 8$

---

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

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

$$
= x
$$

✔ Answer: $x$

---

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

Simplify constants: $\frac{8}{4} = 2$

$$
= 2(y - 16)
$$

✔ Answer: $2(y - 16)$

---

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

---

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

Cancel $(3x - 4)$ (assume $3x \neq 4$), and simplify:

- Numerator: $3x$
- Denominator: $18x^2 = 18x \cdot x$

So:
$$
\frac{3x}{18x^2} = \frac{1}{6x}
$$

✔ Answer: $\frac{1}{6x}$

---

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

Cancel $(x + 5)$ (assume $x \neq -5$):

$$
= x(x - 5)
$$

✔ Answer: $x(x - 5)$

---

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

Simplify:

- $\frac{9}{27} = \frac{1}{3}$
- $\frac{y}{y^2} = \frac{1}{y}$
- Cancel $(y - 1)$ (assume $y \neq 1$)

So:
$$
= \frac{1}{3} \cdot \frac{1}{y} \cdot (2y - 1) = \frac{2y - 1}{3y}
$$

✔ Answer: $\frac{2y - 1}{3y}$

---

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

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

Remaining:
$$
= x(x + 1)(x - 1)
$$

✔ Answer: $x(x + 1)(x - 1)$ or $x(x^2 - 1)$

---

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

Simplify:

- $\frac{8}{12} = \frac{2}{3}$
- $\frac{y}{y^2} = \frac{1}{y}$
- $\frac{(y + 4)^2}{(y + 4)} = (y + 4)$

So:
$$
= \frac{2}{3} \cdot \frac{1}{y} \cdot (y + 4) = \frac{2(y + 4)}{3y}
$$

✔ Answer: $\frac{2(y + 4)}{3y}$

---

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

Cancel $x$ and one $(3x - 2)$ (assume $x \neq 0$, $3x \neq 2$):

Numerator: $1$

Denominator: $7x^2(3x - 2)$

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

✔ Answer: $\frac{1}{7x^2(3x - 2)}$

---

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

Simplify:

- $\frac{3}{18} = \frac{1}{6}$
- $\frac{x^3}{x^4} = \frac{1}{x}$
- $\frac{(5y - 3)}{(5y - 3)^3} = \frac{1}{(5y - 3)^2}$
- $(y + 3)$ remains

So:
$$
= \frac{1}{6} \cdot \frac{1}{x} \cdot \frac{1}{(5y - 3)^2} \cdot (y + 3) = \frac{y + 3}{6x(5y - 3)^2}
$$

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

---

✔ Section B: Simplify the following algebraic fractions

Now we factor and cancel.

---

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

Factor numerator: $4(2x + 1)$

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

✔ Answer: $2(2x + 1)$

---

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

Factor: $2(y + 3)$

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

✔ Answer: $\frac{y + 3}{2}$

---

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

Factor denominator: $7(2x - 3)$

$$
= \frac{7x}{7(2x - 3)} = \frac{x}{2x - 3}
$$

✔ Answer: $\frac{x}{2x - 3}$

---

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

Factor denominator: $3y(1 + 9y)$

Numerator: $9y^2 = 9y \cdot y$

$$
= \frac{9y^2}{3y(1 + 9y)} = \frac{3y}{1 + 9y}
$$

✔ Answer: $\frac{3y}{1 + 9y}$

---

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

Factor denominator: $5(x - 4)$

$$
= \frac{x - 4}{5(x - 4)} = \frac{1}{5}, \quad x \neq 4
$$

✔ Answer: $\frac{1}{5}$

---

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

Factor numerator: $6y(1 - 5y)$

Denominator: $24y^2$

$$
= \frac{6y(1 - 5y)}{24y^2} = \frac{1 - 5y}{4y}
$$

✔ Answer: $\frac{1 - 5y}{4y}$

---

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

Note: $4 - x = -(x - 4)$

$$
= \frac{x - 4}{-(x - 4)} = -1, \quad x \neq 4
$$

✔ Answer: $-1$

---

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

Factor numerator: $3(7 - x)$

Denominator: $6x(7 - x)$

Wait: $42x - 6x^2 = -6x^2 + 42x = -6x(x - 7)$

But $7 - x = -(x - 7)$, so:

Numerator: $3(7 - x) = -3(x - 7)$

Denominator: $-6x(x - 7)$

So:
$$
= \frac{-3(x - 7)}{-6x(x - 7)} = \frac{3}{6x} = \frac{1}{2x}, \quad x \neq 7
$$

✔ Answer: $\frac{1}{2x}$

---

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

Factor denominator: $x^2 + 7x + 10 = (x + 2)(x + 5)$

$$
= \frac{x + 2}{(x + 2)(x + 5)} = \frac{1}{x + 5}, \quad x \neq -2
$$

✔ Answer: $\frac{1}{x + 5}$

---

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

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

$$
= \frac{(x - 4)(x - 9)}{x - 4} = x - 9, \quad x \neq 4
$$

✔ Answer: $x - 9$

---

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

Factor numerator: $x^2 - 8x - 20 = (x - 10)(x + 2)$

Denominator: $9x + 18 = 9(x + 2)$

$$
= \frac{(x - 10)(x + 2)}{9(x + 2)} = \frac{x - 10}{9}, \quad x \neq -2
$$

✔ Answer: $\frac{x - 10}{9}$

---

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

Numerator: $5(x + 8)$

Denominator: $x^2 + 6x - 16 = (x + 8)(x - 2)$

$$
= \frac{5(x + 8)}{(x + 8)(x - 2)} = \frac{5}{x - 2}, \quad x \neq -8
$$

✔ Answer: $\frac{5}{x - 2}$

---

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

Numerator: $4(3x + 5)$

Denominator: $9x^2 + 9x - 10$

Try factoring: find two numbers that multiply to $9 \times (-10) = -90$, add to $9$: $15$ and $-6$

So:
$$
9x^2 + 15x - 6x - 10 = 3x(3x + 5) -2(3x + 5) = (3x - 2)(3x + 5)
$$

So:
$$
= \frac{4(3x + 5)}{(3x - 2)(3x + 5)} = \frac{4}{3x - 2}, \quad x \neq -\frac{5}{3}
$$

✔ Answer: $\frac{4}{3x - 2}$

---

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

Numerator: $(x + 2)(x + 3)$

Denominator: $(x + 12)(x + 2)$

$$
= \frac{(x + 2)(x + 3)}{(x + 12)(x + 2)} = \frac{x + 3}{x + 12}, \quad x \neq -2
$$

✔ Answer: $\frac{x + 3}{x + 12}$

---

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

Factor numerator: $x^2 - 7x - 44 = (x - 11)(x + 4)$

Denominator: $x^2 - 17x + 66 = (x - 11)(x - 6)$

$$
= \frac{(x - 11)(x + 4)}{(x - 11)(x - 6)} = \frac{x + 4}{x - 6}, \quad x \neq 11
$$

✔ Answer: $\frac{x + 4}{x - 6}$

---

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

Factor numerator: $6x^2 - x - 1 = (3x + 1)(2x - 1)$

Check: $3x \cdot 2x = 6x^2$, $3x(-1) + 1(2x) = -3x + 2x = -x$, $1 \cdot (-1) = -1$. Correct.

Denominator: $15x^2 + 8x + 1$

Find two numbers: $15 \cdot 1 = 15$, need sum 8 → $5$ and $3$

So: $15x^2 + 5x + 3x + 1 = 5x(3x + 1) + 1(3x + 1) = (5x + 1)(3x + 1)$

So:
$$
= \frac{(3x + 1)(2x - 1)}{(5x + 1)(3x + 1)} = \frac{2x - 1}{5x + 1}, \quad x \neq -\frac{1}{3}
$$

✔ Answer: $\frac{2x - 1}{5x + 1}$

---

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

Numerator: difference of squares: $(x - y)(x + y)$

Denominator: $(x + y)^2$

$$
= \frac{(x - y)(x + y)}{(x + y)^2} = \frac{x - y}{x + y}, \quad x \neq -y
$$

✔ Answer: $\frac{x - y}{x + y}$

---

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

Numerator: $4y^2 - 9x^2 = (2y)^2 - (3x)^2 = (2y - 3x)(2y + 3x)$

Denominator: $4x^2y + 6x^3 = 2x^2(2y + 3x)$

So:
$$
= \frac{(2y - 3x)(2y + 3x)}{2x^2(2y + 3x)} = \frac{2y - 3x}{2x^2}, \quad 2y + 3x \neq 0
$$

✔ Answer: $\frac{2y - 3x}{2x^2}$

---

✔ 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. $2(2x + 1)$
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. $\frac{2y - 3x}{2x^2}$

---

Let me know if you'd like this in a printable format or with step-by-step explanations for any specific problem!