Expand (i) (a - b) (a + b), (ii) (a - b) (a² + ab + b²) and (iii) (a - b)(a³ + a²b + ab² + b³), Do you see a pattern? Class 8
easyExpand (i) (a - b) (a + b), (ii) (a - b) (a² + ab + b²) and (iii) (a - b)(a³ + a²b + ab² + b³), Do you see a pattern? Class 8
Question 1.
Expand (i) (a - b) (a + b), (ii) (a - b) (a2 + ab + b2) and (iii) (a - b)(a3 + a2b + ab2 + b3), Do you see a pattern? What would he the next identity in the pattern that you see? Can you check it by expanding? Class 8
Solution:
(i) (a - b) (a + b) = a(a + b) - b (a + b)
= a2 + ab - ba - b2 = a2 - b2
(ii) (a - b) (a2 + ab + b2)
= a (a2 + ab + b2) - b(a2 + ab + b2)
= a3 + a2b + ab2 - a2b - ab2 - b3
= a3 - b3
(iii) (a - b)(a3 + a3b + ab2 + b3)
= a (a3 + a2b + ab2 + b3) - b(a3 - a2b + ab2 + b3)
= a4 + a3b + a2b2 + ab3 - a3b - a2b2 - ab3 - b4
= a4 - b4
Yes, there is pattern, according to which
(a - b) (a4 + a3b + a2b2 + ab3 + b4)
= a5 — b5
Because, (a - b) (a4 + a3b + a2b2 + ab3 + b4) = a (a4 + a3b + a2b2 + ab3 + b4) - b (a4 + a3b + a2b2 + ab3 + b4)
= a5 + a4b + a3b2 + a2b3 + ab4 - a4b - a3b2 - a2b3 - ab4 - b5
= a5 - b5