Perform the operation(s) and simplify completely.

((8x^3-12x^2+18x-27)/(2x-3))-x^2+4x

1 answer

To simplify this expression, we will start by dividing the polynomials:

First, divide the numerator (8x^3 - 12x^2 + 18x - 27) by the denominator (2x - 3). This can be done using polynomial long division:

4x^2 + 3x + 9
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2x - 3 | 8x^3 - 12x^2 + 18x - 27
- (8x^3 - 12x^2 + 18x - 27)
___________________________
0

As we can see, there is no remainder, so the quotient is 4x^2 + 3x + 9.

Now, we can rewrite the given expression:

(4x^2 + 3x + 9) - x^2 + 4x

Next, combine like terms:

4x^2 + 3x + 9 - x^2 + 4x

Simplifying further:

(4x^2 - x^2) + (3x + 4x) + 9

3x^2 + 7x + 9

Therefore, the simplified form of ((8x^3-12x^2+18x-27)/(2x-3))-x^2+4x is 3x^2 + 7x + 9.