Simplify completely
(8x^3-12x^2+18x-27)/(2x-3)-x^2+4x
First, let's simplify the fraction (8x^3-12x^2+18x-27)/(2x-3):
Using polynomial long division, we divide 8x^3-12x^2+18x-27 by 2x-3:
4x^2 - 1
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2x - 3 | 8x^3 - 12x^2 + 18x - 27
- (8x^3 - 12x^2)
________________
0 - 0 + 18x - 27
- (0 - 0 + 18x - 27)
________________
0
Therefore, (8x^3-12x^2+18x-27)/(2x-3) simplifies to 4x^2 - 1.
Now, we can rewrite the expression as (4x^2 - 1) - x^2 + 4x:
Combining like terms, we have 4x^2 - x^2 + 4x - 1:
This simplifies to 3x^2 + 4x - 1.
Therefore, the simplified form of (8x^3-12x^2+18x-27)/(2x-3)-x^2+4x is 3x^2 + 4x - 1.
Is that the correct answer?
Apologies for the mistake in my previous response. Let's redo the simplification.
We have the expression (8x^3-12x^2+18x-27)/(2x-3)-x^2+4x.
First, let's simplify the fraction (8x^3-12x^2+18x-27)/(2x-3) using polynomial long division:
4x^2 + 2x + 8
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2x - 3 | 8x^3 - 12x^2 + 18x - 27
- (8x^3 - 12x^2)
____________________
0 + 30x - 27
- (0 - 6x + 9)
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36x - 36
- (36x - 54)
____________________
18
Therefore, (8x^3-12x^2+18x-27)/(2x-3) simplifies to 4x^2 + 2x + 8.
Now, we can rewrite the expression as (4x^2 + 2x + 8) - x^2 + 4x:
Combining like terms, we have 3x^2 + 6x + 8.
Therefore, the simplified form of (8x^3-12x^2+18x-27)/(2x-3)-x^2+4x is 3x^2 + 6x + 8.
Are you guessing? What made you think your first response wasn’t correct?
Apologies for the confusion caused by the incorrect first response.
To clarify, let's simplify the expression again:
(8x^3 - 12x^2 + 18x - 27)/(2x - 3) - x^2 + 4x
First, let's simplify the fraction (8x^3 - 12x^2 + 18x - 27)/(2x - 3):
Using polynomial long division or synthetic division, we find that the quotient is 4x^2 - 2x - 9, with a remainder of 0. Therefore, (8x^3 - 12x^2 + 18x - 27)/(2x - 3) simplifies to 4x^2 - 2x - 9.
Now, let's rewrite the expression as 4x^2 - 2x - 9 - x^2 + 4x:
Combining like terms, we have 3x^2 + 2x - 9.
Therefore, the simplified form of (8x^3 - 12x^2 + 18x - 27)/(2x - 3) - x^2 + 4x is 3x^2 + 2x - 9.
To simplify the expression (8x^3 - 12x^2 + 18x - 27)/(2x - 3) - x^2 + 4x, we need to find a common denominator and combine like terms.
Step 1: Start by simplifying the division (8x^3 - 12x^2 + 18x - 27)/(2x - 3).
Since the numerator is a cubic expression and the denominator is a linear expression, we can use long division to simplify it:
```
4x^2 -2x +4
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2x - 3 | 8x^3 -12x^2 +18x -27
- (8x^3 -12x^2)
_______________
0 +18x -27
- (18x - 27)
_______________
0
```
Therefore, the division simplifies to: 4x^2 - 2x + 4.
Step 2: Substitute the simplified division back into the original equation.
(8x^3 - 12x^2 + 18x - 27)/(2x - 3) - x^2 + 4x
= (4x^2 - 2x + 4) - x^2 + 4x
Step 3: Combine like terms.
= 4x^2 - 2x + 4 - x^2 + 4x
= (4x^2 - x^2) + (-2x + 4x) + (4 + 4)
= 3x^2 + 2x + 8
Therefore, the completely simplified form of the expression is 3x^2 + 2x + 8.
To simplify the given expression: (8x^3 - 12x^2 + 18x - 27) / (2x - 3) - x^2 + 4x, we need to simplify each part and then combine like terms.
First, let's simplify (8x^3 - 12x^2 + 18x - 27) / (2x - 3) using polynomial division or long division. Here's how you can do it step by step:
1. Divide the first term of the dividend (8x^3) by the first term of the divisor (2x), which gives you 4x^2.
2. Multiply the entire divisor (2x - 3) by the result from step 1 (4x^2) and subtract it from the dividend.
(8x^3 - 12x^2 + 18x - 27) - (4x^2 * (2x - 3))
Simplifying the subtraction, we get: 8x^3 - 12x^2 + 18x - 27 - (8x^3 - 12x^2)
Combining like terms, we have: 8x^3 - 12x^2 + 18x - 27 - 8x^3 + 12x^2
This simplifies to: 6x - 27.
3. Repeat the process with the simplified dividend (6x - 27) and the divisor (2x - 3).
Divide the first term of the simplified dividend (6x) by the first term of the divisor (2x), which gives you 3.
4. Multiply the entire divisor (2x - 3) by the result from step 3 (3) and subtract it from the simplified dividend.
(6x - 27) - (3 * (2x - 3))
Simplifying the subtraction, we get: 6x - 27 - (6x - 9)
Combining like terms, we have: 6x - 27 - 6x + 9
This simplifies to: -18.
So, the expression (8x^3 - 12x^2 + 18x - 27) / (2x - 3) simplifies to 3 - 18, which further simplifies to -15.
Now, let's simplify the remaining terms, -x^2 + 4x:
The expression doesn't have any like terms to combine, so it remains the same.
Finally, we combine the simplified terms from both parts of the expression:
-15 + (-x^2 + 4x) (Combining like terms)
-15 - x^2 + 4x (Rearranging the terms)
Therefore, the completely simplified form of the given expression is -x^2 + 4x - 15.