1. **Problem statement:** Factorise the expressions: ⑦ $8a^4 - 64a$ ⑧ $-x^3 - 27$ ⑨ $343 - (a - 1)^3$ --- 2. **Recall formulas:** - Difference of cubes: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ - Sum of cubes: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ - Common factor extraction: factor out the greatest common factor (GCF). --- 3. **Factorise ⑦ $8a^4 - 64a$:** - Extract GCF: $8a$ - Inside parentheses: $a^3 - 8$ - Recognize $a^3 - 8$ as difference of cubes with $a^3$ and $2^3$ $$8a^4 - 64a = 8a(a^3 - 8)$$ $$= 8a (a - 2)(a^2 + 2a + 4)$$ --- 4. **Factorise ⑧ $-x^3 - 27$:** - Rewrite as $-(x^3 + 27)$ - Recognize $x^3 + 27$ as sum of cubes with $x^3$ and $3^3$ $$-x^3 - 27 = -(x^3 + 27) = -(x + 3)(x^2 - 3x + 9)$$ --- 5. **Factorise ⑨ $343 - (a - 1)^3$:** - Recognize $343 = 7^3$ - Expression is difference of cubes: $7^3 - (a - 1)^3$ $$343 - (a - 1)^3 = (7 - (a - 1)) (7^2 + 7(a - 1) + (a - 1)^2)$$ Simplify first factor: $$7 - (a - 1) = 7 - a + 1 = 8 - a$$ Simplify second factor: $$7^2 = 49$$ $$7(a - 1) = 7a - 7$$ $$(a - 1)^2 = a^2 - 2a + 1$$ Sum: $$49 + 7a - 7 + a^2 - 2a + 1 = a^2 + (7a - 2a) + (49 - 7 + 1) = a^2 + 5a + 43$$ Final factorisation: $$343 - (a - 1)^3 = (8 - a)(a^2 + 5a + 43)$$