Interactive worksheet for simplifying rational functions with step-by-step problems.

Let me work through each of the four rational function simplification problems step by step.

Problem 1 (Blue): (2x-6)/(4x-12)

Factor numerator: 2x - 6 = 2(x - 3)
Factor denominator: 4x - 12 = 4(x - 3)

So: (2x-6)/(4x-12) = 2(x-3)/4(x-3)

Cancel (x-3): = 2/4 = 1/2

Problem 2 (Yellow): (3x²-27)/(3x²-9x)

Factor numerator: 3x² - 27 = 3(x² - 9) = 3(x-3)(x+3)
Factor denominator: 3x² - 9x = 3x(x - 3)

So: (3x²-27)/(3x²-9x) = 3(x-3)(x+3)/[3x(x-3)]

Cancel 3 and (x-3): = (x+3)/x

Problem 3 (Pink): (x²+4x+3)/(x²-2x-15)

Factor numerator: x² + 4x + 3 = (x+1)(x+3)
Factor denominator: x² - 2x - 15 = (x-5)(x+3)

So: (x²+4x+3)/(x²-2x-15) = (x+1)(x+3)/[(x-5)(x+3)]

Cancel (x+3): = (x+1)/(x-5)

Problem 4 (Green): (x²-4)/(x²-4x+4)

Factor numerator: x² - 4 = (x-2)(x+2) [difference of squares]
Factor denominator: x² - 4x + 4 = (x-2)(x-2) = (x-2)²

So: (x²-4)/(x²-4x+4) = (x-2)(x+2)/[(x-2)(x-2)]

Cancel one (x-2): = (x+2)/(x-2)

Let me verify all answers:

1. (2x-6)/(4x-12): Test x=5 → (10-6)/(20-12) = 4/8 = 1/2 ✓
2. (3x²-27)/(3x²-9x): Test x=4 → (48-27)/(48-36) = 21/12 = 7/4; (x+3)/x = 7/4 ✓
3. (x²+4x+3)/(x²-2x-15): Test x=6 → (36+24+3)/(36-12-15) = 63/9 = 7; (x+1)/(x-5) = 7/1 = 7 ✓
4. (x²-4)/(x²-4x+4): Test x=5 → (25-4)/(25-20+4) = 21/9 = 7/3; (x+2)/(x-2) = 7/3 ✓

All verified!

Final Answer:
1. (2x-6)/(4x-12) = 2(x-3)/4(x-3) = 2/4 = 1/2
2. (3x²-27)/(3x²-9x) = 3(x-3)(x+3)/[3x(x-3)] = (x+3)/x
3. (x²+4x+3)/(x²-2x-15) = (x+1)(x+3)/[(x-5)(x+3)] = (x+1)/(x-5)
4. (x²-4)/(x²-4x+4) = (x-2)(x+2)/[(x-2)(x-2)] = (x+2)/(x-2)