Simple division problem I am not grasping. z^3-125 / z-5

Question

I know the answer is z^2 + 5z + 25 but where does the 5z come from? 

Answer 1

Steve,

 

Try long division and/or synthetic division (Google or Khan Academy if you don't know). Also, I'm finding it hard to explain either in this format. (I prefer long division, but they are the same.)

 

Long division:

 

z³ - 125 ÷ z - 5 same as

 

z -5 divided into z³ + 0z² + 0z - 125

 

gives

 

z²      ->  z³ + 0z² - (z³ - 5z²) = 5z² + 0

 

which leads to

 

5z      -> 5z² + 0z - (5z² - 25z) = 25z - 125

 

which leads to

 

25     -> 25z - 125 - (25z - 125) = 0

 

Answer: z² + 5z + 25

Answer 2

z³ - 125 = z³ - 5³ is a "difference of cubes" of the form a³ - b^3.  A difference of cubes has a standard factors of:

 

a³ - b³ = (a-b)(a²+ab+b²)

 

For your problem, a = z and b = 5.  Hence:

 

z³ - 5³  = (z-5)(z² + 5z + 25)

 

Hence:

 

 z³-5³        (z-5)(z²+5z+25)

--------  =  ----------------------  =  z² + 5z + 25

 (z-5)              (z-5)