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# Factors of Binomials: Sum And Difference of Two Cubes

**Factors of Binomials : Sum And Difference of Two Cubes**

**Sum of Cubes : X^3 + Y^3 = (X + Y) (X^2 - XY + Y^2)**

**Difference of Cubes : X^3 – Y^3 = (X – Y ) (X^2 + XY + Y^2)**

**What are Cubes ?**

**Cubes are numbers which has exact cube roots. For example 27 is a cube, it has an exact cube root which is 3 since (3)(3)(3) = 27.**

**Example One : Factor 8X^3 + 27Y^3**

**8X^3 + 27Y^3 = (2X)^3 + (3Y)^3**

** = (2X + 3Y ) { (2X)^2 -(2X)(3Y) + (3Y)^2}**

** = (2X + 3Y ) ( 4X^2 - 6XY + 9Y^2)**

**Example Two : Factor 27a^3 - 64b^6**

**27a^3 – 64b^6 = (3a)^3 - (4b^2)^3 **

** = (3a -4b^2) {(3a^2 + (3a)(4b^2) + (4b^2)^2**

** = (3a – 4b^2) (9a^2 + 12 ab^2 + 16b^4**

**Example Three : Factor X^9 – Y^9**

**X^9 – Y^9 = ( X^3)^3 - (Y^3 )^3**

** = (X^3 - Y^3 ) { (X^3)^2 + X^3Y^3 + (y^3)^2}**

** = (X – Y ) (X^2 + XY + Y^2) (X^6 + X^3Y^3 + Y^6)**

**Example Four : Factor (2x – Y)^3 - 8 = (2X – Y )^3 – 2^3**

** = { (2X –Y) – 2 } {(2X – Y)^2 + 2(2X- Y ) + 4 }**

**Example Five : Factor 343X^12 - Y ^9**

**343X^12 – Y^9 = (7X^4)^3 - ( Y^3)^3**

** = (7X^4 - Y^3) { (7X^4)^2 + (7X^4) (Y^3) + (Y^3)^2 }**

** = (7X^4 – Y^3 ) (49 X^8 + 7X^4 Y^3 + Y^6 )**

**SOURCE : COLLEGE ALGEBRA By**

** REES**

** SPARKS**

** REES **