Question

Find the product:

(За – 4b – c) (9a² + 16b² + c² + 12ab – 4bc + 3ca)

Answer 1

Given expression: (За – 4b – c)(9a² + 16b² + c² + 12ab – 4bc + 3ca)

Step-wise calculation:

Multiply each term in the first bracket by each term in the second bracket:

(3a)(9a² + 16b² + c² + 12ab – 4bc + 3ca) – (4b)(9a² + 16b² + c² + 12ab – 4bc + 3ca) – (c)(9a² + 16b² + c² + 12ab – 4bc + 3ca)

Calculate each part:

1. 3a × 9a² = 27a³

2. 3a × 16b² = 48ab²

3. 3a × c² = 3ac²

4. 3a × 12ab = 36a²b

5. 3a × (–4bc) = –12abc

6. 3a × 3ca = 9a²c

7. –4b × 9a² = –36a²b

8. –4b × 16b² = –64b³

9. –4b × c² = –4bc²

10. –4b × 12ab = –48ab²

11. –4b × (–4bc) = 16b²c

12. –4b × 3ca = –12abc

13. –c × 9a² = –9a²c

14. –c × 16b² = –16b²c

15. –c × c² = –c³

16. –c × 12ab = –12abc

17. –c × (–4bc) = 4bc²

18. –c × 3ca = –3ac²

Now add all these results:

27a³ + 48ab² + 3ac² + 36a²b – 12abc + 9a²c – 36a²b – 64b³ – 4bc² – 48ab² + 16b²c – 12abc – 9a²c – 16b²c – c³ – 12abc + 4bc² – 3ac²

Group like terms:

• (a³): (27a³)

• (b³): (–64b³)

• (c³): (–c³)

• (a²b): (36a²b – 36a²b = 0)

• (a²c): (9a²c – 9a²c = 0)

• (ab²): (48ab² – 48ab² = 0)

• (bc²): (–4bc² + 4bc² = 0)

• (b²c): (16b²c – 16b²c = 0)

• (ac²): (3ac² – 3ac² = 0)

• (abc): (–12abc – 12abc – 12abc = –36abc)

27a³ – 64b³ – c³ – 36abc