Worksheet for simplifying rational expressions in elementary algebra.

Let’s solve each problem one by one. We’ll simplify rational expressions by factoring numerators and denominators, then canceling common factors.

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1) $\frac{35u}{35u^2}$

- Both numerator and denominator have 35 and $u$.
- Cancel 35: $\frac{u}{u^2}$
- Cancel one $u$: $\frac{1}{u}$

✔ Final Answer: $\boxed{\frac{1}{u}}$

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2) $\frac{45v^2}{25x}$

- 45 and 25 share a common factor of 5.
- $\frac{45}{25} = \frac{9}{5}$
- No common variables (v² and x are different)
- So: $\frac{9v^2}{5x}$

✔ Final Answer: $\boxed{\frac{9v^2}{5x}}$

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3) $\frac{x - 8}{x^2 + x - 72}$

- Factor denominator: Find two numbers that multiply to -72 and add to +1 → 9 and -8
- So: $x^2 + x - 72 = (x + 9)(x - 8)$
- Expression becomes: $\frac{x - 8}{(x + 9)(x - 8)}$
- Cancel $(x - 8)$: $\frac{1}{x + 9}$

✔ Final Answer: $\boxed{\frac{1}{x + 9}}$

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4) $\frac{p^2 - 3p - 54}{p - 9}$

- Factor numerator: Two numbers that multiply to -54 and add to -3 → -9 and +6
- So: $p^2 - 3p - 54 = (p - 9)(p + 6)$
- Expression: $\frac{(p - 9)(p + 6)}{p - 9}$
- Cancel $(p - 9)$: $p + 6$

✔ Final Answer: $\boxed{p + 6}$

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5) $\frac{56v - 72}{32v}$

- Factor numerator: GCF of 56 and 72 is 8 → $8(7v - 9)$
- Denominator: $32v = 8 \cdot 4v$
- So: $\frac{8(7v - 9)}{8 \cdot 4v} = \frac{7v - 9}{4v}$

✔ Final Answer: $\boxed{\frac{7v - 9}{4v}}$

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6) $\frac{a + 7}{a^2 + 6a - 7}$

- Factor denominator: Two numbers that multiply to -7 and add to +6 → +7 and -1
- So: $a^2 + 6a - 7 = (a + 7)(a - 1)$
- Expression: $\frac{a + 7}{(a + 7)(a - 1)}$
- Cancel $(a + 7)$: $\frac{1}{a - 1}$

✔ Final Answer: $\boxed{\frac{1}{a - 1}}$

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7) $\frac{5v^2 - 5v}{1 - v}$

- Factor numerator: $5v(v - 1)$
- Note: $1 - v = -(v - 1)$
- So: $\frac{5v(v - 1)}{-(v - 1)} = -5v$

✔ Final Answer: $\boxed{-5v}$

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8) $\frac{2 - x}{x^2 + 4x - 12}$

- Numerator: $2 - x = -(x - 2)$
- Denominator: Factor → two numbers that multiply to -12 and add to +4 → +6 and -2
- So: $x^2 + 4x - 12 = (x + 6)(x - 2)$
- Expression: $\frac{-(x - 2)}{(x + 6)(x - 2)} = \frac{-1}{x + 6}$

✔ Final Answer: $\boxed{-\frac{1}{x + 6}}$

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9) $\frac{v^2 + 7v - 30}{9v^2 + 90v}$

- Numerator: Factor → two numbers that multiply to -30 and add to +7 → +10 and -3
→ $(v + 10)(v - 3)$
- Denominator: Factor out 9v → $9v(v + 10)$
- Expression: $\frac{(v + 10)(v - 3)}{9v(v + 10)} = \frac{v - 3}{9v}$

✔ Final Answer: $\boxed{\frac{v - 3}{9v}}$

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10) $\frac{b^2 + b - 30}{3b^2 + 18b}$

- Numerator: Factor → two numbers that multiply to -30 and add to +1 → +6 and -5
→ $(b + 6)(b - 5)$
- Denominator: Factor out 3b → $3b(b + 6)$
- Expression: $\frac{(b + 6)(b - 5)}{3b(b + 6)} = \frac{b - 5}{3b}$

✔ Final Answer: $\boxed{\frac{b - 5}{3b}}$

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11) $\frac{3x^2 + 5x - 2}{7x^2 + 12x - 4}$

- Numerator: Use AC method or trial → factors of 3*(-2)=-6 that add to +5 → +6 and -1
→ Split middle term: $3x^2 + 6x - x - 2 = 3x(x + 2) -1(x + 2) = (3x - 1)(x + 2)$
- Denominator: 7*(-4)=-28, find factors that add to +12 → +14 and -2
→ $7x^2 + 14x - 2x - 4 = 7x(x + 2) -2(x + 2) = (7x - 2)(x + 2)$
- Expression: $\frac{(3x - 1)(x + 2)}{(7x - 2)(x + 2)} = \frac{3x - 1}{7x - 2}$

✔ Final Answer: $\boxed{\frac{3x - 1}{7x - 2}}$

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12) $\frac{3b^2 + 26b - 9}{5b^2 + 40b - 45}$

- Numerator: 3*(-9)=-27, factors that add to +26 → +27 and -1
→ $3b^2 + 27b - b - 9 = 3b(b + 9) -1(b + 9) = (3b - 1)(b + 9)$
- Denominator: Factor out 5 first? Wait — let’s factor directly:
5b² + 40b - 45 → factor out 5: $5(b^2 + 8b - 9)$
Then factor inside: b² + 8b - 9 → (b + 9)(b - 1)
So denominator: $5(b + 9)(b - 1)$
- Expression: $\frac{(3b - 1)(b + 9)}{5(b + 9)(b - 1)} = \frac{3b - 1}{5(b - 1)}$

✔ Final Answer: $\boxed{\frac{3b - 1}{5(b - 1)}}$

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13) $\frac{3a^2 + 21a - 90}{3a^2 + 31a + 10}$

- Numerator: Factor out 3 → $3(a^2 + 7a - 30)$
Factor inside: a² + 7a - 30 → (a + 10)(a - 3)
So numerator: $3(a + 10)(a - 3)$
- Denominator: 3a² + 31a + 10 → factors of 3*10=30 that add to 31 → 30 and 1
→ 3a² + 30a + a + 10 = 3a(a + 10) +1(a + 10) = (3a + 1)(a + 10)
- Expression: $\frac{3(a + 10)(a - 3)}{(3a + 1)(a + 10)} = \frac{3(a - 3)}{3a + 1}$

✔ Final Answer: $\boxed{\frac{3(a - 3)}{3a + 1}}$

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14) $\frac{5m^3 - 57mn + 70n^2}{2m^2 - 16mn - 40n^2}$

Wait — this looks like it might be miswritten. The numerator has m³ but no m² term? Let me check if it's supposed to be quadratic in m.

Actually, looking again — probably typo? But assuming it’s correct as written:

Numerator: $5m^3 - 57mn + 70n^2$ — this is cubic in m, hard to factor without more context.

But wait — maybe it’s meant to be $5m^2 - 57mn + 70n^2$? That would make sense with denominator being quadratic.

Looking at the pattern of other problems, likely a typo. Let’s assume it’s:

$\frac{5m^2 - 57mn + 70n^2}{2m^2 - 16mn - 40n^2}$

Now factor both.

Numerator: 5m² - 57mn + 70n²

Find factors of 5*70=350 that add to -57 → -50 and -7

Split: 5m² - 50mn - 7mn + 70n² = 5m(m - 10n) -7n(m - 10n) = (5m - 7n)(m - 10n)

Denominator: 2m² - 16mn - 40n² → factor out 2: 2(m² - 8mn - 20n²)

Factor inside: m² - 8mn - 20n² → factors of -20 that add to -8 → -10 and +2

→ (m - 10n)(m + 2n)

So denominator: 2(m - 10n)(m + 2n)

Expression: $\frac{(5m - 7n)(m - 10n)}{2(m - 10n)(m + 2n)} = \frac{5m - 7n}{2(m + 2n)}$

✔ Final Answer: $\boxed{\frac{5m - 7n}{2(m + 2n)}}$

*(Note: If original was indeed cubic, we’d need more info — but based on context, quadratic makes sense.)*

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15) $\frac{v^3 + 11v^2 + 18v}{v^2 + v - 2}$

- Numerator: Factor out v → $v(v^2 + 11v + 18)$
Factor quadratic: v² + 11v + 18 → (v + 2)(v + 9)
So numerator: $v(v + 2)(v + 9)$
- Denominator: v² + v - 2 → (v + 2)(v - 1)
- Expression: $\frac{v(v + 2)(v + 9)}{(v + 2)(v - 1)} = \frac{v(v + 9)}{v - 1}$

✔ Final Answer: $\boxed{\frac{v(v + 9)}{v - 1}}$

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16) $\frac{2x^3 + 16x^2 + 24x}{x^2 - x - 6}$

- Numerator: Factor out 2x → $2x(x^2 + 8x + 12)$
Factor quadratic: x² + 8x + 12 → (x + 2)(x + 6)
So numerator: $2x(x + 2)(x + 6)$
- Denominator: x² - x - 6 → (x - 3)(x + 2)
- Expression: $\frac{2x(x + 2)(x + 6)}{(x - 3)(x + 2)} = \frac{2x(x + 6)}{x - 3}$

✔ Final Answer: $\boxed{\frac{2x(x + 6)}{x - 3}}$

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17) $\frac{xy + 3x - 2y - 6}{y^2 + y - 6}$

- Numerator: Group terms → (xy + 3x) + (-2y - 6) = x(y + 3) -2(y + 3) = (x - 2)(y + 3)
- Denominator: y² + y - 6 → (y + 3)(y - 2)
- Expression: $\frac{(x - 2)(y + 3)}{(y + 3)(y - 2)} = \frac{x - 2}{y - 2}$

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

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18) $\frac{ax - ay + bx - by}{ax - ay - bx + by}$

- Numerator: Group → (ax - ay) + (bx - by) = a(x - y) + b(x - y) = (a + b)(x - y)
- Denominator: (ax - ay) - (bx - by) = a(x - y) - b(x - y) = (a - b)(x - y)
- Expression: $\frac{(a + b)(x - y)}{(a - b)(x - y)} = \frac{a + b}{a - b}$

✔ Final Answer: $\boxed{\frac{a + b}{a - b}}$

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Final Answers Summary:

1) $\boxed{\frac{1}{u}}$
2) $\boxed{\frac{9v^2}{5x}}$
3) $\boxed{\frac{1}{x + 9}}$
4) $\boxed{p + 6}$
5) $\boxed{\frac{7v - 9}{4v}}$
6) $\boxed{\frac{1}{a - 1}}$
7) $\boxed{-5v}$
8) $\boxed{-\frac{1}{x + 6}}$
9) $\boxed{\frac{v - 3}{9v}}$
10) $\boxed{\frac{b - 5}{3b}}$
11) $\boxed{\frac{3x - 1}{7x - 2}}$
12) $\boxed{\frac{3b - 1}{5(b - 1)}}$
13) $\boxed{\frac{3(a - 3)}{3a + 1}}$
14) $\boxed{\frac{5m - 7n}{2(m + 2n)}}$
15) $\boxed{\frac{v(v + 9)}{v - 1}}$
16) $\boxed{\frac{2x(x + 6)}{x - 3}}$
17) $\boxed{\frac{x - 2}{y - 2}}$
18) $\boxed{\frac{a + b}{a - b}}$