What is the result when 2x3−9x2+11x−6 is divided by x−3 ? Show your work.
To divide the polynomial 2x^3 - 9x^2 + 11x - 6 by x - 3, we can use long division.
We start by writing the dividend (2x^3 - 9x^2 + 11x - 6) and the divisor (x - 3) and then begin the division process:
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x - 3 | 2x^3 - 9x^2 + 11x - 6
We divide the first term in the dividend (2x^3) by the first term in the divisor (x) which gives us 2x^2. We then place this result above the line, aligned with the variable terms:
2x^2
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x - 3 | 2x^3 - 9x^2 + 11x - 6
Next, we multiply the divisor (x - 3) by the obtained quotient (2x^2):
2x^2
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x - 3 | 2x^3 - 9x^2 + 11x - 6
- 2x^3 + 6x^2
We subtract this result from the dividend and bring down the next term (-9x^2):
2x^2
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x - 3 | 2x^3 - 9x^2 + 11x - 6
- 2x^3 + 6x^2
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-15x^2 + 11x - 6
Since the degree of the new dividend (-15x^2 + 11x - 6) is less than the divisor (x - 3), there are no more x^2 terms to divide. We now move on to the next term, which is -15x^2.
We divide -15x^2 by x, which gives us -15x. We place this quotient above the line, aligned with the variable terms:
2x^2 - 15x
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x - 3 | 2x^3 - 9x^2 + 11x - 6
- 2x^3 + 6x^2
____________
-15x^2 + 11x - 6
Next, we multiply the divisor (x - 3) by the obtained quotient (-15x):
2x^2 - 15x
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x - 3 | 2x^3 - 9x^2 + 11x - 6
- 2x^3 + 6x^2
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-15x^2 + 45x
-15x^2 + 45x
We subtract this result from the previous dividend (-15x^2 + 11x - 6) and bring down the next term (11x):
2x^2 - 15x
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x - 3 | 2x^3 - 9x^2 + 11x - 6
- 2x^3 + 6x^2
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-15x^2 + 11x - 6
-15x^2 + 45x
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-34x - 6
The new dividend (-34x - 6) has a degree less than the divisor, so we move to the next term. We divide -34x by x, giving us -34. We place this quotient above the line, aligned with the variable terms:
2x^2 - 15x - 34
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x - 3 | 2x^3 - 9x^2 + 11x - 6
- 2x^3 + 6x^2
____________
-15x^2 + 11x - 6
-15x^2 + 45x
____________
-34x - 6
Next, we multiply the divisor (x - 3) by the obtained quotient (-34):
2x^2 - 15x - 34
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x - 3 | 2x^3 - 9x^2 + 11 - 6
- 2x^3 + 6x^2
____________
-15x^2 + 11x - 6
-15x^2 + 45x
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-34x - 6
-34x + 102
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-108
We subtract this result from the previous dividend and have a remainder of -108. Therefore, the result when 2x^3 - 9x^2 + 11x - 6 is divided by x - 3 is 2x^2 - 15x - 34 with a remainder of -108.
Final Result:
(2x^2 - 15x - 34) - (108 / (x - 3))