Question 4 of 27advanced🔍 AnalyzeLong Answer3 marks

Expand (i) (a−b)(a+b)(a - b)(a + b), (ii) (a−b)(a2+ab+b2)(a - b)(a^2 + ab + b^2) and (iii) (a−b)(a3+a2b+ab2+b3)(a - b)(a^3 + a^2b + ab^2 + b^3). Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

(i)
Answer

a2−b2a^2 - b^2

(ii)
Answer

a3−b3a^3 - b^3

(iii)
Answer

a4−b4a^4 - b^4

Explanation

(i) Expand: a2+ab−ba−b2=a2−b2a^2 + ab - ba - b^2 = a^2 - b^2 (middle terms cancel) (ii) Expand: a3+a2b+ab2−a2b−ab2−b3=a3−b3a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 = a^3 - b^3 (iii) Expand: a4+a3b+a2b2+ab3−a3b−a2b2−ab3−b4=a4−b4a^4 + a^3b + a^2b^2 + ab^3 - a^3b - a^2b^2 - ab^3 - b^4 = a^4 - b^4

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