(2x^(3)-x^(2)-24x+12)+(2x-1) divide.
To divide the expression (2x^3 - x^2 - 24x + 12) + (2x-1), we can rewrite it as a fraction, with the numerator being the given expression and the denominator being the divisor (2x-1):
(2x^3 - x^2 - 24x + 12) / (2x-1)
To divide polynomials, we can use polynomial long division. Here is the step-by-step process:
x^2 + 10x + 2
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2x - 1 | 2x^3 - x^2 - 24x + 12
1. Divide the first term of the numerator (2x^3) by the first term of the denominator (2x). The result is x^2.
x^2
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2x - 1 | 2x^3 - x^2 - 24x + 12
2. Multiply the denominator (2x-1) by the quotient (x^2). The result is 2x^3 - x^2.
x^2 + 10x + 2
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2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
______________
- 24x + 12
3. Subtract the result of the multiplication from the numerator.
- 24x + 12
x^2 + 10x + 2
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2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
______________
- 24x + 12
4. Repeat steps 1-3 with the new result.
4. Divide the first term of the new resulting polynomial (-24x) by the first term of the denominator (2x). The result is -12.
x^2 + 10x + 2 - 12
____________________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
______________
- 24x + 12
- (- 24x + 12)
_______________
0
5. Multiply the denominator (2x-1) by the new quotient (-12). The result is -24x + 12.
x^2 + 10x + 2 - 12
____________________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
______________
- 24x + 12
- (- 24x + 12)
_______________
0
6. Subtract the result of the multiplication from the previous result. The remainder is 0.
x^2 + 10x + 2 - 12
____________________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
______________
- 24x + 12
- (- 24x + 12)
_______________
0
7. Write the final result as the quotient plus the remainder divided by the divisor:
x^2 + 10x + 2 - 12 / (2x - 1)
Simplifying the expression x^2 + 10x + 2 - 12, we get:
x^2 + 10x - 10
Therefore, the division of (2x^3 - x^2 - 24x + 12) + (2x-1) is x^2 + 10x - 10.