✎ Problem:
Factor the following:
- 8x⁵y⁷ - 20x³y⁶
- 18y⁴ + 30y² - 42y
- x⁴ - 1
- 81 - p²
- y² - 20y + 100
- x² + 24x + 144
- y³ - 1/27
- 27x⁶ - y⁶
✎ Answers:
For Numbers 1 and 2, this involves factoring the common monomial factor which is the common factor, as a product of numerical and literal coefficients all terms in a given polynomial have in common, such that the common factor can be evenly divided into each term.
- 1. 8x⁵y⁷ - 20x³y⁶ = 4x³y⁶ (2x²y - 5)
- 2. 18y⁴ + 30y² - 42y = 6y (3y³ + 5y - 7)
For Numbers 3 and 4, this involves factoring the difference of two squares, which is in the form a² - b² = (a + b)(a - b)
- 3. x⁴ - 1 = (x² + 1)(x² - 1)
- 4. 81 - p² = (9 + p)(9 - p)
For Numbers 5 and 6, this involves factoring the perfect square trinomial, which is in the form:
- a² + 2ab + b² = (a + b)²
- a² - 2ab + b² = (a - b)²
So:
- 5. y² - 20y + 100 = (y - 10)²
- 6. x² + 24x + 144 = (x + 12)²
For Numbers 7 and 8, this involves factoring the sum or difference of two cubes, which is in the form:
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
So:
- 7. y³ - 1/27 = (y - 1/3)(y² + y/3 + 1/9)
- 8. 27x⁶ - y⁶ = (3x² - y²)(9x³ + 3x²y² + y³)
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