Perform the operation(s) and simplify completely.
((8x^3-12x^2+18x-27)/(2x-3))-x^2+4x
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
___________________________
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.
To simplify the given expression, let's start by simplifying the numerator of the fraction.
The numerator is 8x^3 - 12x^2 + 18x - 27. We can factor out a GCF (Greatest Common Factor) of 3 from each term:
3(8x^3/3 - 12x^2/3 + 18x/3 - 27/3)
This simplifies to:
3(8/3 x^3 - 4x^2 + 6x - 9)
Now let's simplify the denominator, which is 2x - 3.
The complete expression becomes:
3(8/3 x^3 - 4x^2 + 6x - 9) / (2x - 3) - x^2 + 4x
Next, let's work on the division:
3(8/3 x^3 - 4x^2 + 6x - 9) / (2x - 3) becomes:
(24/3 x^3 - 12x^2 + 18x - 27) / (2x - 3)
Simplifying the division, we get:
8x^3/1 - 4x^2/1 + 6x/1 - 27/1 / (2x - 3)
Now, let's focus on the division of each term:
8x^3 / 2x = 4x^2
- 4x^2 / 2x = -2x
6x / 2x = 3
-27 / 2x = -27/2x
So, the expression becomes:
4x^2 - 2x + 3 - 27/2x - x^2 + 4x
Combining like terms, we have:
- x^2 + 4x^2 - 2x + 4x + 3 - 27/2x
This simplifies to:
3x^2 + 2x + 3 - 27/2x
Therefore, the simplified expression is 3x^2 + 2x + 3 - 27/2x.
To simplify the given expression: ((8x^3-12x^2+18x-27)/(2x-3))-x^2+4x, we'll follow these steps:
Step 1: Simplify the fraction
Step 2: Combine like terms
Let's begin Step 1 by simplifying the fraction:
To divide polynomials, we'll use polynomial long division. Here's how it works:
4x^2 + (22x + 34)
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2x - 3 | 8x^3 - 12x^2 + 18x - 27
Step 1a: Divide the first term of the dividend by the first term of the divisor:
4x^2
Step 1b: Multiply the entire divisor (2x - 3) by the term we found in Step 1a:
4x^2 * (2x - 3) = 8x^3 - 12x^2
Step 1c: Subtract the result we obtained in Step 1b from the original dividend:
(8x^3 - 12x^2 + 18x - 27) - (8x^3 - 12x^2) = 18x - 27
Step 1d: Bring down the next term from the original dividend:
18x - 27
Step 1e: Repeat Steps 1a to 1d until we have a remainder of zero.
Now, we can rewrite the given expression with the simplified fraction:
((4x^2 + 22x + 34) / (2x - 3)) - x^2 + 4x
Moving to Step 2, let's combine like terms:
First, let's focus on the fraction portion ((4x^2 + 22x + 34) / (2x - 3)).
There are no like terms to combine, so we'll rewrite it as is.
Now, let's combine the like terms outside the fraction:
- x^2 + 4x
Finally, we can rewrite the completely simplified expression as:
(4x^2 + 22x + 34) / (2x - 3) - x^2 + 4x
That's the final simplified form of the expression!