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Factors of Binomials: Sum And Difference of Two Cubes

Updated on November 9, 2011

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

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    • cristina327 profile image
      Author

      cristina327 5 years ago from Manila

      HI ronhi great to hear from you again. Thank you for dropping by and appreciating this hub. It is so amusing to know that you are "standing out student" in Mathematics. Mathematics is always source of fun for me. I enjoy so much its workings. You can also enjoy it. Blessings to you always. Best regards.

    • cristina327 profile image
      Author

      cristina327 5 years ago from Manila

      Hi lord de cross great to hear from you again. Thanks for paying a visit. I am greatly encouraged by your visit and comments. May you be blessed today and always. Best regards.

    • ronhi profile image

      ronhi 5 years ago from Kenya

      you are making me love mathematics...and yet i was a "standing out student" in maths; Literally- i used to stand outside the class when the lessons were going on

    • Lord De Cross profile image

      Joseph De Cross 5 years ago

      Excellent Christina!

      Now I can easily factorized my bills that got tripled literally. Nice class for our kids. Voting up!

      LORD