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