Perform the operation(s) and simplify completely.
(8x3−12x2+18x−27)2x−3 − x2+4x
(1 point)
x2
x
+9
To simplify the expression (8x^3 - 12x^2 + 18x - 27) / (2x - 3) - x^2 + 4x, we need to perform the division and combine like terms.
Step 1: Divide the polynomial (8x^3 - 12x^2 + 18x - 27) by (2x - 3)
Using polynomial long division, we have:
4x^2 + 2x + 3
__________________
2x - 3 | 8x^3 - 12x^2 + 18x - 27
- (8x^3 - 12x^2)
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0x^2 + 18x
- (0x^2 - 0x)
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18x - 27
- (18x - 27)
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0
The result of the division is 4x^2 + 2x + 3.
Step 2: Simplify the expression by combining like terms
Combining the simplified form of the division (4x^2 + 2x + 3) with the rest of the expression - x^2 + 4x, we get:
(4x^2 + 2x + 3) - x^2 + 4x
Simplifying further by combining like terms:
4x^2 - x^2 + 2x + 4x + 3
Resulting in:
3x^2 + 6x + 3
To perform the operation and simplify completely, we need to first expand (8x^3 - 12x^2 + 18x - 27) and then distribute the expression (2x - 3) to it. Finally, we will combine like terms and simplify.
Step 1: Expanding the expression:
(8x^3 - 12x^2 + 18x - 27) * (2x - 3)
Multiplying each term of the first expression by the terms in the second expression, we get:
(8x^3 * 2x) - (8x^3 * 3) - (12x^2 * 2x) + (12x^2 * 3) + (18x * 2x) - (18x * 3) - (27 * 2x) + (27 * 3)
Simplifying, we get:
16x^4 - 24x^3 - 24x^3 + 36x^2 + 36x^2 - 54x - 54x + 81
Step 2: Combining like terms:
16x^4 - 48x^3 + 72x^2 - 108x + 81
Step 3: Simplifying further:
The expression is a quadratic equation in terms of x, so we can attempt to factor it or divide it using synthetic division to find any possible roots. However, in this case, the expression doesn't factor easily and doesn't have any rational roots. So, we can't simplify it any further.
Therefore, the simplified expression is:
16x^4 - 48x^3 + 72x^2 - 108x + 81
To simplify the expression, we will first perform the operation inside the parentheses.
(8x^3 - 12x^2 + 18x - 27)(2x - 3) - x^2 + 4x
Now we can distribute the first term, (8x^3 - 12x^2 + 18x - 27), to each term inside the second parentheses.
16x^4 - 24x^3 + 36x^2 - 54x - 24x^2 + 36x - 54 - x^2 + 4x
Now we can combine like terms.
16x^4 - 24x^3 + (36x^2 - 24x^2 - x^2) + (36x + 4x) - 54 - 54
= 16x^4 - 24x^3 + 11x^2 + 40x - 108
Therefore, the simplified expression is:
16x^4 - 24x^3 + 11x^2 + 40x - 108.