The volume of a rectangular prism is represented by the expression x^ + 7x^2 + 15x + 9. the height is represented by the expression ( x + 1 ). if the width and the height are the same, what is the width of the prism?

Width of the prism is ##4## units.

As the volume of a rectangular prism, whose length is ##l##, height is ##h## and width is ##w## is ##lxxhxxw##.

As the volume of rectangular prism is ##x^3+7x^2+15x+9##,

and height is ##(x+1)## and width and height being same, height too is ##(x+1)##

we can have its length by dividing ##x^3+7x^2+15x+9## by ##(x+1)(x_1)=x^2+2x+1##.

Dividing ##x^3+7x^2+15x+9## by ##(x^2+2x+1)##,

##x(x^2+2x+1)+5(x^2+2x+1)+4x+4##

But as volume is ##lxxhxxw##, ##4x+4=4(x+1)## too should be a multiple of ##x^2+2x+1=(x+1)^2##,

which is possible if ##x+1=4## i.e. ##x=3##

Hence width is ##4## and height too is ##4##

Note that volume is ##3^3+7xx3^2+15xx3+9=27+63+45+9=144##

and length is ##144/(4xx4)=9##.