EXERCISE 1.13 (a) Factorise: \( \begin{array}{lll}\text { (1) } 27 x^{3}-1 & \text { (2) } 8 x^{3}+1 & \text { (3) } 64 x^{3}-y^{3} \\ \text { (4) } 125-729 x^{3} & \text { (5) } a^{3} b^{3}-216 & \text { (6) } \\ \text { (60) } \\ \text { (7) } 8 a^{4}-64 a & \text { (8) }-x^{3}-27 & \text { (9) }\end{array} \)
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Let's dive into these exercises and factor the expressions! 1. **27x³ - 1**: This is a difference of cubes, which can be factored using the formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, let \( a = 3x \) and \( b = 1 \): \[ 27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1) \] 2. **8x³ + 1**: This is a sum of cubes, factored using \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Let \( a = 2x \) and \( b = 1 \): \[ 8x^3 + 1 = (2x + 1)(4x^2 - 2x + 1) \] 3. **64x³ - y³**: Again, this is a difference of cubes with \( a = 4x \) and \( b = y \): \[ 64x^3 - y^3 = (4x - y)(16x^2 + 4xy + y^2) \] 4. **125 - 729x³**: This expression can also be viewed as a difference of cubes. Let \( a = 5 \) and \( b = 9x \): \[ 125 - 729x^3 = (5 - 9x)(25 + 45x + 81x^2) \] 5. **a³b³ - 216**: Noticing that \( 216 = 6^3 \), we can express this as a difference of cubes with \( a = ab \) and \( b = 6 \): \[ a^3b^3 - 216 = (ab - 6)(a^2b^2 + 6ab + 36) \] 6. **8a⁴ - 64a**: This expression has a common factor of \( 8a \): \[ 8a^4 - 64a = 8a(a^3 - 8) = 8a(a - 2)(a^2 + 2a + 4) \] 7. **-x³ - 27**: This can be rewritten as \(- (x^3 + 27)\). Recognizing it as a sum of cubes with \( a = x \) and \( b = 3 \): \[ -x^3 - 27 = -(x + 3)(x^2 - 3x + 9) \] And voilà! Each expression has been factored accordingly! If you have more questions or need further help, feel free to ask!
