Question

How do you divide `(9x^3+5)div(2x-3)` using long division?

Answer 1

`9/2 x^2 + 27/4 x + 81/8 + (1947/72)/(2 x - 3)`

Explanation:

First, let's get rid of coefficients and multiply them in later. This yields
`(x^3 + 5/9)/(x-3/2)`

Long division tasks us with asking how many times things go into each other evenly. With polynomials, we just begin with their leading terms.

For this instance, we start with `x^2`, since `x` goes into `x^3` `x^2` times. That means what is left over is
`R = x^3 + 5/9 - x^2(x- 3/2) = 3/2 x^2 + 5/9`
And therefore, our next value is `3/2 x`, since `x` goes into `3x^2` that many times.
`R' = 3/2 x^2 + 5/9 - 3/2 x (x - 3/2) = 9/4 x + 5/9`
For our final step, we have `9/4` for the same reasons as above:
`R'' = 9/4 x + 5/9 - 9/4 (x - 3/2) = 5/9 + 27/8 = 273/72`

Therefore, multiplying back in the `9/2` from the beginning, our final answer is
`9/2 x^2 + 27/4 x + 81/8 + (1947/72)/(2x-3)`