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Difference of Two Squares,
Sum of Two Cubes and other Properties


When it comes to multiplying out terms and expressions with brackets involved. As well as the associative, commutative and distributive properties shown here.

There are other standard properties/rules that are handy to be aware of in Algebra.

They apply to some situations when there are square or cubes involved.






Properties Involving Squares



Difference of Two Squares:

  x^2 \space {\text{--}} \space y^2 \space = \space ( x + y )( x \space {\text{--}} \space y )

  x^2 - 16 \space\space = \space\space x^2 - 4^2 \space\space = \space ( x + 4 )( x - 4 )

  ( x + 2 )( x - 2 ) \space\space = \space\space x^2 - 2^2 \space\space = \space\space x^2 - 4



Square of a Difference:

  ( x \space {\text{--}} \space y )^2 \space\space = \space\space x^2 \space {\text{--}} \space 2xy + y^2

  ( x - 2 )^2 \space\space = \space\space x^2 - 2x(2) + 2^2

= \space x^2 - 4x + 4



Square of a Sum:

  ( x + y )^2 \space\space = \space\space x^2 + 2xy + y^2

  ( x + 4 )^2 \space\space = \space\space x^2 + 2x(4) + 4^2

= \space x^2 + 8x + 16






Properties Involving Cubes:



Cube of a Sum & Cube of a Difference:

  ( x + y )^3 \space\space = \space\space x^3 + 3x^2y + 3xy^2 + y^3

  ( x \space {\text{--}} \space y )^3 \space\space = \space\space x^3 \space {\text{--}} \space 3x^2y + 3xy^2 \space {\text{--}} \space y^3

  ( 3x + 2 )^3

= \space (3x)^3 + 3(3x)^2(2) + 3(3x)(2)^2 + 2^3

= \space\space 27x^3 + 18^2 + 36x + 8


  ( 2x - 4 )^3

= \space (2x)^3 - 3(2x)^2({\text{-}}4) + 3(2x)({\text{-}}4)^2 - ({\text{-}}4)^3

= \space\space 8x^3 + 48^2 + 96x + 64




Sum of Two Cubes, Difference of Two Cubes:

  x^3 + y^3 \space\space = \space\space ( x + y )( x^2 \space {\text{--}} \space xy + y^2 )

  x^3 \space {\text{--}} \space y^3 \space\space = \space\space ( x + y )( x^2 + xy + y^2 )

  x^3 + 8 \space\space = \space\space x^3 + 2^3

= \space ( x + 2 )( x^2 - 2x + 4 )


  x^3 - 64 \space\space = \space\space x^3 - 4^3

= \space ( x - 4 )( x^2 + 4x + 16 )






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