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

Question

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

UpStudy AI · Accepted Answer

Factor the expression by following steps:
- step0: Factor:
  $$27x^{3}-1$$
- 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 $$ 5 x^{3}+40 $$.
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 $$ -x^{3}-27 $$.
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 $$ \frac{1}{27} x^{3}+216 $$.
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 $$ 125-729 x^{3 $$.
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 $$ 8 x^{3}+1 $$.
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 $$ 64 x^{3}-y^{3 $$.
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 $$ 8 a^{4}-64 a $$.
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 $$ a^{3} b^{3}-\frac{1}{8 $$.
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. **$$27 x^{3}-1$$**:
   $$
   (3x-1)(9x^{2}+3x+1)
   $$

2. **$$8 x^{3}+1$$**:
   $$
   (2x+1)(4x^{2}-2x+1)
   $$

3. **$$64 x^{3}-y^{3}$$**:
   $$
   (4x-y)(16x^{2}+4xy+y^{2})
   $$

4. **$$125-729 x^{3}$$**:
   $$
   (5-9x)(25+45x+81x^{2})
   $$

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

6. **$$5 x^{3}+40$$**:
   $$
   5(x+2)(x^{2}-2x+4)
   $$

7. **$$8 a^{4}-64 a$$**:
   $$
   8a(a-2)(a^{2}+2a+4)
   $$

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 result could be found for this input.

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

If you need further assistance or explanations on any of these factorizations, feel free to ask!
Here are the factorizations for the given expressions: 1. $$27 x^{3}-1 = (3x-1)(9x^{2}+3x+1)$$ 2. $$8 x^{3}+1 = (2x+1)(4x^{2}-2x+1)$$ 3. $$64 x^{3}-y^{3} = (4x-y)(16x^{2}+4xy+y^{2})$$ 4. $$125-729 x^{3} = (5-9x)(25+45x+81x^{2})$$ 5. $$a^{3} b^{3}-\frac{1}{8} = \frac{1}{8}(2ab-1)(4a^{2}b^{2}+2ab+1)$$ 6. $$5 x^{3}+40 = 5(x+2)(x^{2}-2x+4)$$ 7. $$8 a^{4}-64 a = 8a(a-2)(a^{2}+2a+4)$$ 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 more help with any of these, let me know!

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