What is the other factor of the binomial X cubed minus 8 if one factor is X minus 2?

To find the other factor of the polynomial ( x^3 - 8 ) given that one factor is ( x - 2 ), we can perform polynomial long division or synthetic division. Since ( x^3 - 8 ) can be rewritten using the difference of cubes, we recognize it can be factored as:

[

x^3 - 8 = (x - 2)(x^2 + 2x + 4)

]

Here, ( x^3 - 8 ) is the difference of cubes, where ( 8 ) is ( 2^3 ). According to the formula for factoring the difference of cubes, ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ), we can set ( a = x ) and ( b = 2 ).

When we apply this formula, we recognize that:

• ( a - b ) gives us ( x - 2 )

• ( a^2 + ab + b^2 ) translates to ( x^2 + 2x + 4 ), as ( b^2 = 2^2 ) equals 4