Perform the operation(s) and simplify completely.
(8x3−12x2+18x−27)2x−3 − x2+4x
First, we will simplify the expression (8x^3 - 12x^2 + 18x - 27) / (2x - 3):
(8x^3 - 12x^2 + 18x - 27) / (2x - 3)
Factor out common terms in the numerator:
[2(4x^3 - 6x^2 + 9x - 13)] / (2x - 3)
Notice that we can factor out a 2 from the numerator as well:
2(4x^3 - 6x^2 + 9x - 13) / (2x - 3)
Now, simplify the expression x^2 + 4x:
x^2 + 4x
Factor out x:
x(x + 4)
Combining the two simplified expressions, we have:
2(4x^3 - 6x^2 + 9x - 13) / (2x - 3) - x(x + 4)
To perform the given operation and simplify completely, follow these steps:
Step 1: Simplify the expression within parentheses.
The expression (8x^3−12x^2+18x−27) can be simplified as:
8x^3 − 12x^2 + 18x − 27.
Step 2: Expand the expression (8x^3−12x^2+18x−27) squared.
To square the expression, we multiply each term by itself. The squared expression becomes:
(8x^3 − 12x^2 + 18x − 27)2 = (8x^3 − 12x^2 + 18x − 27)(8x^3 − 12x^2 + 18x − 27).
Step 3: Use the FOIL method to multiply the expressions.
FOIL stands for First, Outer, Inner, Last. Multiply the terms in each group accordingly:
(8x^3 − 12x^2 + 18x − 27)(8x^3 − 12x^2 + 18x − 27)
= (8x^3)(8x^3) + (8x^3)(-12x^2) + (8x^3)(18x) + (8x^3)(-27) +
(-12x^2)(8x^3) + (-12x^2)(-12x^2) + (-12x^2)(18x) + (-12x^2)(-27) +
(18x)(8x^3) + (18x)(-12x^2) + (18x)(18x) + (18x)(-27) +
(-27)(8x^3) + (-27)(-12x^2) + (-27)(18x) + (-27)(-27).
Step 4: Simplify the terms.
Now we simplify the terms from the multiplication:
64x^6 - 96x^5 + 144x^4 - 216x^3 - 96x^5 + 144x^4 - 216x^3 + 324x^2 + 144x^4 - 216x^3 + 324x^2 - 486x - 216x^3 + 324x^2 - 486x + 729
= 64x^6 - 192x^5 + 432x^4 - 864x^3 + 648x^2 - 972x + 729.
Step 5: Simplify the remaining terms of the expression.
The remaining term is -x^2 + 4x.
Step 6: Combine the simplified terms.
Combine the simplified terms of the expression we have so far:
64x^6 - 192x^5 + 432x^4 - 864x^3 + 648x^2 - 972x + 729 - x^2 + 4x.
The simplified expression after combining like terms is:
64x^6 - 192x^5 + 432x^4 - 865x^3 + 647x^2 - 968x + 729.
To perform the given operation and simplify completely, let's break it down step by step.
Step 1: Expand the expression
Begin by expanding the expression (8x^3 − 12x^2 + 18x − 27)2x^−3:
(8x^3 − 12x^2 + 18x − 27)2x^−3 − x^2 + 4x
= 8x^3(2x^−3) − 12x^2(2x^−3) + 18x(2x^−3) − 27(2x^−3) − x^2 + 4x
Now, simplify the exponents:
= 8x^3 * (2/x^3) − 12x^2 * (2/x^3) + 18x * (2/x^3) − 27 * (2/x^3) − x^2 + 4x
= 16/x^3 + 24/x − 36x/x^3 − 54/x^3 − x^2 + 4x
Step 2: Combine like terms
Next, combine the like terms by adding or subtracting:
= (16 − 36 − 54)/x^3 + (24 − 36)x/x − x^2 + 4x
= -74/x^3 − 12x − x^2 + 4x
Step 3: Rearrange the terms
Rearrange the terms to put them in descending order:
= -74/x^3 − x^2 − 8x + 4x
= -74/x^3 − x^2 − 4x
Therefore, the simplified expression is -74/x^3 − x^2 − 4x.