Sum and Difference of Cubes - Finding New Algebraic Identities - Chapter 4 Class 9 - Exploring Algebraic Identities (Ganita Manjari I)

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Sum and Difference of Cubes SUM OF TWO CUBES FORMULA Example: ■(8a^3+27=(2a)^3+3^3@=(2a+3)(4a^2-6a+9) ) DIFFERENCE OF TWO CUBES FORMULA x^3-y^3=(x-y)(x^2+xy+y^2 ) Factors into: (x-y)" and " (x^2+xy+y^2 ) Example: (&27x^3-64=(3x)^3-4^3@=& (3x-4)(9x^2+12x+16) ) 𝒙^𝟑+𝒚^𝟑 𝑭𝒐𝒓𝒎𝒖𝒍𝒂 Our identity is 𝒙^𝟑+𝒚^𝟑=(𝒙+𝒚)(𝒙^𝟐−𝒙𝒚+𝒚^𝟐 ) We can prove this using algebra (𝒙+𝒚)(𝒙^𝟐−𝒙𝒚+𝒚^𝟐 ) =𝑥(𝑥^2−𝑥𝑦+𝑦^2 )+𝑦(𝑥^2−𝑥𝑦+𝑦^2 ) =(𝑥^3−𝑥^2 𝑦+𝑥𝑦^2 )+(𝑥^2 𝑦−𝑥𝑦^2+𝑦^3 ) =𝑥^3−𝑥^2 𝑦+𝑥𝑦^2+𝑥^2 𝑦−𝑥𝑦^2+𝑦^3 =𝒙^𝟑+𝒚^𝟑 𝒙^𝟑−𝒚^𝟑 𝑭𝒐𝒓𝒎𝒖𝒍𝒂 Our identity is 𝒙^𝟑−𝒚^𝟑=(𝒙−𝒚)(𝒙^𝟐+𝒙𝒚+𝒚^𝟐 ) We can prove this using algebra (𝒙−𝒚)(𝒙^𝟐+𝒙𝒚+𝒚^𝟐 ) =𝑥(𝑥^2+𝑥𝑦+𝑦^2 )−𝑦(𝑥^2+𝑥𝑦+𝑦^2 ) =(𝑥^3+𝑥^2 𝑦+𝑥𝑦^2 )−(𝑥^2 𝑦+𝑥𝑦^2+𝑦^3 ) =𝑥^3+𝑥^2 𝑦+𝑥𝑦^2−𝑥^2 𝑦−𝑥𝑦^2−𝑦^3 =𝒙^𝟑−𝒚^𝟑