Question 5 - Figure it out - Page 142, 143 - Chapter 6 Class 8 - We Distribute yet things Multiply (Ganita Prakash)

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Question 5 Expand (i) (a – b) (a + b), (ii) (a – b) (a2 + ab + b2 ), (iii) (a – b)(a3 + a2 b + ab2 + b3 ), Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?Let’s expand one-by-one For (i) (a – b) (a + b) Now, (𝒂−𝒃)(𝒂+𝒃)=𝑎(𝑎+𝑏)−𝑏(𝑎+𝑏) =𝑎^2+𝑎𝑏−𝑏𝑎−𝑏^2 Since ab = ba =𝑎^2+𝑎𝑏−𝑎𝑏−𝑏^2 =𝒂^𝟐−𝒃^𝟐 For (ii) (a – b) (a2 + ab + b2) Now, (𝑎−𝑏)(𝑎^2+𝑎𝑏+𝑏^2 ) =𝑎(𝑎^2+𝑎𝑏+𝑏^2 )−𝑏(𝑎^2+𝑎𝑏+𝑏^2 ) =𝑎 × 𝑎^2+𝑎 × 𝑎𝑏+𝑎𝑏^2−𝑏𝑎^2−𝑏 × 𝑎𝑏−𝑏 × 𝑏^2 =𝐚^𝟑+𝐚^𝟐 𝐛+〖𝐚𝐛〗^𝟐−〖𝐛𝐚〗^𝟐−〖𝐚𝐛〗^𝟐−𝐛^𝟑 =a^3+(a^2 b−a^2 b)+(ab^2−𝑎𝑏^2)−b^3 =𝒂^𝟑−𝒃^𝟑 For (iii) (a – b)(a3 + a2 b + ab2 + b3) Now, (𝑎−𝑏)(𝑎^3+𝑎^2 𝑏+𝑎𝑏^2+𝑏^3 ) =𝑎(𝑎^3+𝑎^2 𝑏+𝑎𝑏^2+𝑏^3 )−𝑏(𝑎^3+𝑎^2 𝑏+𝑎𝑏^2+𝑏^3 ) =𝐚^𝟒+𝐚^𝟑 𝐛+〖𝐚^𝟐 𝐛〗^𝟐+𝐚𝐛^𝟑−〖𝐛𝐚〗^𝟑−〖𝐚^𝟐 𝐛〗^𝟐−𝒂𝒃^𝟑−𝒃^𝟒 =a^4+(a^3 b−a^3 b)+(a^2 b^2−a^2 b^2)+(ab^3−𝑎𝑏^3 )−b^4 =𝒂^𝟒−𝒃^𝟒 Thus, our equations are (i) (𝒂−𝒃)(𝒂+𝒃)=𝒂^𝟐−𝒃^𝟐 (ii) (𝒂−𝒃)(𝒂^𝟐+𝒂𝒃+𝒃^𝟐 )=𝒂^𝟑−𝒃^𝟑 (iii) (𝒂−𝒃)(𝒂^𝟑+𝒂^𝟐 𝒃+𝒂𝒃^𝟐+𝒃^𝟑 )=𝒂^𝟒−𝒃^𝟒 What is the pattern? When you multiply (a - b) by a bracket where the powers of a go down and b go up (all plus signs), the middle terms always cancel out! You are left with just a to the next highest power minus b to the next highest power. Next Identity: The next one would have powers up to 4 in the second bracket, and result in powers of 5. (𝒂−𝒃)(𝒂^𝟒+𝒂^𝟑 𝒃+𝒂^𝟐 𝒃^𝟐+𝒂𝒃^𝟑+𝒃^𝟒 ) =𝒂^𝟓−𝒃^𝟓