FXERCISE: 11 Factorise the following $$ \begin{array}{llll} ext { (1) } 27 x^{3}-1 & ext { (2) } 8 x^{3}+1 & ext { (3) } 64 x^{3}-y^{3} \ ext { (4) } 125-729 x^{3} & ext { (5) } a^{3} b^{3}-\frac{1}{8} & ext { (6) } 5 x^{3}+40\end{array} $$ $$ \begin{array}{llll} ext { (7) } 8 a^{4}-64 a & ext { (8) }-x^{3}-27 & ext { (9) } \frac{1}{27} x^{3}+216 \ ext { (10) } 8-(a-1)^{3} & ext { (11) } x^{3}+\frac{1}{x^{3}} & ext { (12) } x^{3}-\frac{1}{x^{3}}\end{array} $$

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FXERCISE: 11 Factorise the following $$ \begin{array}{llll}	ext { (1) } 27 x^{3}-1 & 	ext { (2) } 8 x^{3}+1 & 	ext { (3) } 64 x^{3}-y^{3} \ 	ext { (4) } 125-729 x^{3} & 	ext { (5) } a^{3} b^{3}-\frac{1}{8} & 	ext { (6) } 5 x^{3}+40\end{array} $$ $$ \begin{array}{llll}	ext { (7) } 8 a^{4}-64 a & 	ext { (8) }-x^{3}-27 & 	ext { (9) } \frac{1}{27} x^{3}+216 \ 	ext { (10) } 8-(a-1)^{3} & 	ext { (11) } x^{3}+\frac{1}{x^{3}} & 	ext { (12) } x^{3}-\frac{1}{x^{3}}\end{array} $$

UpStudy AI · Accepted Answer

Factor the expression by following steps:
- step0: Factor:
  $$64x^{3}-y^{3}$$
- step1: Rewrite the expression:
  $$\left(4x\right)^{3}-y^{3}$$
- step2: Factor the expression:
  $$\left(4x-y\right)\left(\left(4x\right)^{2}+4xy+y^{2}\right)$$
- step3: Evaluate:
  $$\left(4x-y\right)\left(16x^{2}+4xy+y^{2}\right)$$
Factor the expression $$ 27 x^{3}-1 $$.
Factor the expression by following steps:
- step0: Factor:
  $$27x^{3}-1$$
- step1: Rewrite the expression:
  $$\left(3x\right)^{3}-1^{3}$$
- step2: Factor the expression:
  $$\left(3x-1\right)\left(\left(3x\right)^{2}+3x\times 1+1^{2}\right)$$
- step3: Evaluate:
  $$\left(3x-1\right)\left(9x^{2}+3x+1\right)$$
Factor the expression $$ -x^{3}-27 $$.
Factor the expression by following steps:
- step0: Factor:
  $$-x^{3}-27$$
- step1: Factor the expression:
  $$-\left(x^{3}+27\right)$$
- step2: Factor the expression:
  $$-\left(x+3\right)\left(x^{2}-3x+9\right)$$
Factor the expression $$ 8 a^{4}-64 a $$.
Factor the expression by following steps:
- step0: Factor:
  $$8a^{4}-64a$$
- step1: Factor the expression:
  $$8a\left(a^{3}-8\right)$$
- step2: Factor the expression:
  $$8a\left(a-2\right)\left(a^{2}+2a+4\right)$$
Factor the expression $$ 125-729 x^{3 $$.
Factor the expression by following steps:
- step0: Factor:
  $$125-729x^{3}$$
- step1: Rewrite the expression:
  $$5^{3}-\left(9x\right)^{3}$$
- step2: Factor the expression:
  $$\left(5-9x\right)\left(5^{2}+5\times 9x+\left(9x\right)^{2}\right)$$
- step3: Evaluate:
  $$\left(5-9x\right)\left(25+45x+81x^{2}\right)$$
Factor the expression $$ \frac{1}{27} x^{3}+216 $$.
Factor the expression by following steps:
- step0: Factor:
  $$\frac{1}{27}x^{3}+216$$
- step1: Factor the expression:
  $$\frac{1}{27}\left(x^{3}+5832\right)$$
- step2: Factor the expression:
  $$\frac{1}{27}\left(x+18\right)\left(x^{2}-18x+324\right)$$
Factor the expression $$ 8 x^{3}+1 $$.
Factor the expression by following steps:
- step0: Factor:
  $$8x^{3}+1$$
- step1: Rewrite the expression:
  $$\left(2x\right)^{3}+1^{3}$$
- step2: Factor the expression:
  $$\left(2x+1\right)\left(\left(2x\right)^{2}-2x\times 1+1^{2}\right)$$
- step3: Evaluate:
  $$\left(2x+1\right)\left(4x^{2}-2x+1\right)$$
Factor the expression $$ 5 x^{3}+40 $$.
Factor the expression by following steps:
- step0: Factor:
  $$5x^{3}+40$$
- step1: Factor the expression:
  $$5\left(x^{3}+8\right)$$
- step2: Factor the expression:
  $$5\left(x+2\right)\left(x^{2}-2x+4\right)$$
Factor the expression $$ a^{3} b^{3}-\frac{1}{8 $$.
Factor the expression by following steps:
- step0: Factor:
  $$a^{3}b^{3}-\frac{1}{8}$$
- step1: Factor the expression:
  $$\frac{1}{8}\left(8a^{3}b^{3}-1\right)$$
- step2: Factor the expression:
  $$\frac{1}{8}\left(2ab-1\right)\left(4a^{2}b^{2}+2ab+1\right)$$
Factor the expression $$ 8-(a-1)^{3 $$.
Factor the expression by following steps:
- step0: Factor:
  $$8-\left(a-1\right)^{3}$$
- step1: Rewrite the expression:
  $$2^{3}-\left(a-1\right)^{3}$$
- step2: Factor the expression:
  $$\left(2-\left(a-1\right)\right)\left(2^{2}+2\left(a-1\right)+\left(a-1\right)^{2}\right)$$
- step3: Evaluate:
  $$\left(2-\left(a-1\right)\right)\left(4+2\left(a-1\right)+\left(a-1\right)^{2}\right)$$
- step4: Factor the expression:
  $$\left(2-a+1\right)\left(4+2\left(a-1\right)+\left(a-1\right)^{2}\right)$$
- step5: Calculate:
  $$\left(3-a\right)\left(4+2\left(a-1\right)+\left(a-1\right)^{2}\right)$$
- step6: Calculate:
  $$\left(3-a\right)\left(3+a^{2}\right)$$
Here are the factorizations for the given expressions:

1. **$$ 27 x^{3}-1 $$**:
   - This is a difference of cubes, which can be factored as:
   $$
   (3x-1)(9x^{2}+3x+1)
   $$

2. **$$ 8 x^{3}+1 $$**:
   - This is a sum of cubes, which can be factored as:
   $$
   (2x+1)(4x^{2}-2x+1)
   $$

3. **$$ 64 x^{3}-y^{3} $$**:
   - This is also a difference of cubes, factored as:
   $$
   (4x-y)(16x^{2}+4xy+y^{2})
   $$

4. **$$ 125-729 x^{3} $$**:
   - This is a difference of cubes, factored as:
   $$
   (5-9x)(25+45x+81x^{2})
   $$

5. **$$ a^{3} b^{3}-\frac{1}{8} $$**:
   - This can be factored as:
   $$
   \frac{1}{8}(2ab-1)(4a^{2}b^{2}+2ab+1)
   $$

6. **$$ 5 x^{3}+40 $$**:
   - This can be factored as:
   $$
   5(x+2)(x^{2}-2x+4)
   $$

7. **$$ 8 a^{4}-64 a $$**:
   - This can be factored as:
   $$
   8a(a-2)(a^{2}+2a+4)
   $$

8. **$$ -x^{3}-27 $$**:
   - This can be factored as:
   $$
   -(x+3)(x^{2}-3x+9)
   $$

9. **$$ \frac{1}{27} x^{3}+216 $$**:
   - This can be factored as:
   $$
   \frac{1}{27}(x+18)(x^{2}-18x+324)
   $$

10. **$$ 8-(a-1)^{3} $$**:
    - This can be factored as:
    $$
    (3-a)(3+a^{2})
    $$

11. **$$ x^{3}+\frac{1}{x^{3}} $$**:
    - No result could be found for this expression.

12. **$$ x^{3}-\frac{1}{x^{3}} $$**:
    - No result could be found for this expression.

If you need further assistance with the last two expressions or any other questions, feel free to ask!
Here are the factorizations for the given expressions: 1. $$ 27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1) $$ 2. $$ 8x^3 + 1 = (2x + 1)(4x^2 - 2x + 1) $$ 3. $$ 64x^3 - y^3 = (4x - y)(16x^2 + 4xy + y^2) $$ 4. $$ 125 - 729x^3 = (5 - 9x)(25 + 45x + 81x^2) $$ 5. $$ a^3b^3 - \frac{1}{8} = \frac{1}{8}(2ab - 1)(4a^2b^2 + 2ab + 1) $$ 6. $$ 5x^3 + 40 = 5(x + 2)(x^2 - 2x + 4) $$ 7. $$ 8a^4 - 64a = 8a(a - 2)(a^2 + 2a + 4) $$ 8. $$ -x^3 - 27 = -(x + 3)(x^2 - 3x + 9) $$ 9. $$ \frac{1}{27}x^3 + 216 = \frac{1}{27}(x + 18)(x^2 - 18x + 324) $$ 10. $$ 8 - (a - 1)^3 = (3 - a)(3 + a^2) $$ 11. $$ x^3 + \frac{1}{x^3} $$ – No factorization found. 12. $$ x^3 - \frac{1}{x^3} $$ – No factorization found. If you need help with the last two expressions or any other questions, feel free to ask!

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