📐 Algebraic Fractions - Complete Guide

Key Principle: Factorize the numerator and denominator, then cancel common factors.

Simplify: (6x² + 9x) / (3x)

Solution:

1. Factor the numerator: 6x² + 9x = 3x(2x + 3)
2. Write as: [3x(2x + 3)] / 3x
3. Cancel 3x from top and bottom
4. Answer: 2x + 3

Simplify: (x² - 5x + 6) / (x - 2)

Solution:

1. Factor the numerator: x² - 5x + 6 = (x - 2)(x - 3)
2. Write as: [(x - 2)(x - 3)] / (x - 2)
3. Cancel (x - 2) from top and bottom
4. Answer: x - 3

Simplify: (x² - 9) / (x² + 6x + 9)

Solution:

1. Factor numerator (difference of squares): x² - 9 = (x + 3)(x - 3)
2. Factor denominator (perfect square): x² + 6x + 9 = (x + 3)(x + 3) = (x + 3)²
3. Write as: [(x + 3)(x - 3)] / [(x + 3)(x + 3)]
4. Cancel one (x + 3)
5. Answer: (x - 3) / (x + 3)

Method: Multiply numerators together and denominators together, then simplify.
Tip: Factor first, then cancel before multiplying!

Calculate: (3x / 4) × (8 / 9x²)

Solution:

1. Write as: (3x × 8) / (4 × 9x²)
2. Simplify: 24x / 36x²
3. Cancel: 24 ÷ 12 = 2, 36 ÷ 12 = 3, x² ÷ x = x
4. Answer: 2 / 3x

Calculate: [(x² - 4) / (x + 3)] × [(x + 3) / (x - 2)]

Solution:

1. Factor x² - 4 = (x + 2)(x - 2)
2. Write as: [(x + 2)(x - 2) / (x + 3)] × [(x + 3) / (x - 2)]
3. Cancel (x + 3) and (x - 2)
4. Answer: x + 2

Method: Keep the first fraction, change ÷ to ×, flip the second fraction (reciprocal).
Remember: "Keep, Change, Flip" (KCF)

Calculate: (2x / 5) ÷ (4x² / 10)

Solution:

1. Keep, Change, Flip: (2x / 5) × (10 / 4x²)
2. Write as: (2x × 10) / (5 × 4x²)
3. Simplify: 20x / 20x²
4. Cancel: Answer: 1 / x

Calculate: [(x² - 1) / (x + 2)] ÷ [(x + 1) / (x² + 3x + 2)]

Solution:

1. Keep, Change, Flip: [(x² - 1) / (x + 2)] × [(x² + 3x + 2) / (x + 1)]
2. Factor: x² - 1 = (x + 1)(x - 1)
3. Factor: x² + 3x + 2 = (x + 1)(x + 2)
4. Write as: [(x + 1)(x - 1) / (x + 2)] × [(x + 1)(x + 2) / (x + 1)]
5. Cancel (x + 2) and (x + 1) twice
6. Answer: x - 1

Method: Find the Lowest Common Denominator (LCD), convert each fraction, then add/subtract numerators.

Steps for Adding/Subtracting:

1. Identify the LCD (factor denominators if needed)
2. Convert each fraction to have the LCD
3. Add or subtract the numerators
4. Simplify if possible

Calculate: (3x / (x + 2)) + (5 / (x + 2))

Solution:

1. Same denominators, so add numerators directly
2. (3x + 5) / (x + 2)
3. Answer: (3x + 5) / (x + 2)

Calculate: (2 / x) + (3 / 4x)

Solution:

1. LCD = 4x
2. Convert first fraction: (2 / x) × (4 / 4) = 8 / 4x
3. Second fraction already has LCD: 3 / 4x
4. Add: (8 + 3) / 4x = 11 / 4x
5. Answer: 11 / 4x

Calculate: (5 / (x + 1)) - (2 / (x - 3))

Solution:

1. LCD = (x + 1)(x - 3)
2. First fraction: [5(x - 3)] / [(x + 1)(x - 3)] = (5x - 15) / [(x + 1)(x - 3)]
3. Second fraction: [2(x + 1)] / [(x + 1)(x - 3)] = (2x + 2) / [(x + 1)(x - 3)]
4. Subtract: (5x - 15 - 2x - 2) / [(x + 1)(x - 3)]
5. Simplify numerator: (3x - 17) / [(x + 1)(x - 3)]
6. Answer: (3x - 17) / [(x + 1)(x - 3)]

Calculate: (3 / (x² - 4)) + (2 / (x + 2))

Solution:

1. Factor x² - 4 = (x + 2)(x - 2)
2. LCD = (x + 2)(x - 2)
3. First fraction already has LCD: 3 / [(x + 2)(x - 2)]
4. Second fraction: [2(x - 2)] / [(x + 2)(x - 2)] = (2x - 4) / [(x + 2)(x - 2)]
5. Add: (3 + 2x - 4) / [(x + 2)(x - 2)]
6. Simplify: (2x - 1) / [(x + 2)(x - 2)]
7. Answer: (2x - 1) / (x² - 4) or (2x - 1) / [(x + 2)(x - 2)]

Calculate: (x / (x² + 5x + 6)) + (2 / (x + 2))

Solution:

1. Factor x² + 5x + 6 = (x + 2)(x + 3)
2. LCD = (x + 2)(x + 3)
3. First fraction already has LCD: x / [(x + 2)(x + 3)]
4. Second fraction: [2(x + 3)] / [(x + 2)(x + 3)] = (2x + 6) / [(x + 2)(x + 3)]
5. Add: (x + 2x + 6) / [(x + 2)(x + 3)]
6. Simplify: (3x + 6) / [(x + 2)(x + 3)]
7. Factor numerator: 3(x + 2) / [(x + 2)(x + 3)]
8. Cancel (x + 2): Answer: 3 / (x + 3)

Strategy: Follow BIDMAS/PEMDAS order. Handle multiplication/division first, then addition/subtraction.

Write as single fraction: (2 / x) × (x / 3) + (1 / 3)

Solution:

1. Do multiplication first: (2x / 3x) = 2 / 3
2. Now add: (2 / 3) + (1 / 3)
3. Same denominators: (2 + 1) / 3
4. Answer: 3 / 3 = 1

Write as single fraction: (3 / (x + 1)) + [(x - 2) / (x + 1)] × [(x + 1) / (x - 2)]

Solution:

1. Do multiplication first: [(x - 2)(x + 1)] / [(x + 1)(x - 2)] = 1
2. Now we have: (3 / (x + 1)) + 1
3. Write 1 with denominator (x + 1): 1 = (x + 1) / (x + 1)
4. Add: [3 + (x + 1)] / (x + 1)
5. Simplify: (x + 4) / (x + 1)
6. Answer: (x + 4) / (x + 1)

Write as single fraction: (1 / x) + (2 / (x + 1)) + (3 / (x² + x))

Solution:

1. Factor x² + x = x(x + 1)
2. LCD = x(x + 1)
3. First: [1(x + 1)] / [x(x + 1)] = (x + 1) / [x(x + 1)]
4. Second: [2x] / [x(x + 1)]
5. Third already has LCD: 3 / [x(x + 1)]
6. Add all: (x + 1 + 2x + 3) / [x(x + 1)]
7. Simplify: (3x + 4) / [x(x + 1)]
8. Answer: (3x + 4) / [x(x + 1)] or (3x + 4) / (x² + x)

❌ Mistake 1: Canceling terms instead of factors

Wrong: (x + 3) / (x + 5) ≠ 3 / 5 (you cannot cancel the x's!)
Remember: Only cancel common FACTORS, not terms

❌ Mistake 2: Adding fractions without common denominator

Wrong: (1 / x) + (1 / y) ≠ 2 / (x + y)
Correct: (1 / x) + (1 / y) = (y + x) / xy

❌ Mistake 3: Forgetting to distribute the negative sign

Example: (3x / 5) - (x + 2) / 5
Wrong: (3x - x + 2) / 5 = (2x + 2) / 5
Correct: (3x - x - 2) / 5 = (2x - 2) / 5

❌ Mistake 4: Forgetting to flip when dividing

Wrong: (a / b) ÷ (c / d) = (ac) / (bd)
Correct: (a / b) ÷ (c / d) = (a / b) × (d / c) = (ad) / (bc)