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

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

• A. X squared plus 2X plus 4
• B. X squared minus 2X plus 4
• C. X squared plus 3X minus 6
• D. X cubed plus 8

Answer: (A)

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

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