volume of a rectangular solid

Question

The volume of a rectangular solid is given by the polynomial 5x−^45x^3−45x^2+45x. The length of the solid is given
by 5x and the width is given by x−3 . Find the height, t, of the solid.

Answer 1

The polynomial for the volume can be factored:

  5x^4 - 5x^3 - 45x^2 + 45x =

  5x ( x^3 - x^2 - 9x + 9) = <--- factors out 5x

  5x ( [x^2 - x^2]  + [-9x + 9] ) <--- factors by grouping

  5x ( x^2(x-1) - 9 (x-1) ) =  <---- factors out x^2 from the first group, -9 from the second group

 5x ( x^2 - 9) (x-1)   <---- factors out the entire quantity (x-1) from both groups

 

 5x ( x + 3)(x-3)(X-1)  <---- x^2 - 9 = (x+3)(x-3) factors as difference of squares pattern

 

So the volume  V = (5x)(x+3)(x-3)(X-1)

 

height = Volume / (length*width)

 

 So when the polynomial for volume is divided by the length and width , 5x and x-3 respectively,

they cancel out.

 

The height is   (x+3)(x-1)