Write the indicated expression as a ratio of two polynomials

Write the indicated expression as a ratio of two polynomials, assuming that

t(x) =

.

=

Solution

t(x) = 4 / ( 3x^3 + 2 )

t(x-1) = 4 / ( 3(x-1)^3 + 2 ) = 4 / ( 3 ( x^3-3x^2+3x-1) + 2 )

t(x-1) = 4 / ( 3x^3 - 9x^2 + 9x -1 )

t(-1) = 4 / 3(-1)^2 + 2 = -4

plugging the values in the expression

t(x-1) - t(-1) / x   = {4 / ( 3x^3 - 9x^2 + 9x -1 ) - (-4)} / x

=

{4 / ( 3x^3 - 9x^2 + 9x -1 ) + 4)} / x

= (12x^3 - 36x^2 + 36x ) / ( 3x^4 - 9x^3 + 9x^2 - x)

= (12x^2 - 36x + 36) / ( 3x^3 - 9x^2 + 9x - 1)