Divide.
(-2x^3+13x^2-19x+6) (-2x+3)
We can use long division or synthetic division to divide (-2x^3+13x^2-19x+6) by (-2x+3).
Using synthetic division:
First, we write (-2x+3) as 3/2*(-4/3x+1), which allows us to use 4/3 as our divisor.
4/3 | -2 13 -19 6
|____ -8 20 -2
-2 5 1 4
Therefore, (-2x^3+13x^2-19x+6) divided by (-2x+3) is equal to -2x^2 + 5x + 1, with a remainder of 4.
So, we can write:
(-2x^3+13x^2-19x+6) = (-2x+3)(-2x^2 + 5x + 1) + 4
To divide the polynomial (-2x^3 + 13x^2 - 19x + 6) by (-2x + 3), we can use polynomial long division. Here are the steps:
Step 1: Set up the long division:
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-2x + 3 | -2x^3 + 13x^2 - 19x + 6
Step 2: Divide the first term (-2x^3) by (-2x), which gives x^2. Write this above the horizontal line.
x^2
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-2x + 3 | -2x^3 + 13x^2 - 19x + 6
Step 3: Multiply (-2x + 3) by x^2 to get (-2x^3 + 3x^2):
x^2
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-2x + 3 | -2x^3 + 13x^2 - 19x + 6
- 2x^3 + 3x^2
Step 4: Subtract (-2x^3 + 3x^2) from (-2x^3 + 13x^2), which gives (0x^3 + 10x^2):
x^2
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-2x + 3 | -2x^3 + 13x^2 - 19x + 6
- 2x^3 + 3x^2
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10x^2 - 19x + 6
Step 5: Bring down the next term (-19x) and repeat steps 2-4.
x^2 + 5x
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-2x + 3 | -2x^3 + 13x^2 - 19x + 6
- 2x^3 + 3x^2
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10x^2 - 19x + 6
- 10x^2 + 15x
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-4x + 6
Step 6: Bring down the next term (+6) and repeat steps 2-4.
x^2 + 5x - 2
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-2x + 3 | -2x^3 + 13x^2 - 19x + 6
- 2x^3 + 3x^2
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10x^2 - 19x + 6
- 10x^2 + 15x
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-4x + 6
- (-4x + 6)
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0
Step 7: Since the remainder is 0, the division is complete. The quotient is x^2 + 5x - 2.
Therefore, the result of dividing (-2x^3 + 13x^2 - 19x + 6) by (-2x + 3) is x^2 + 5x - 2.