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Question 175363: 2x^3 - 5x^2 + 22x + 51/2x + 3
answer: x^2 -4x +17
need a break down of how my text book got that answer i know how they got x^2 and +17 not sure how they got the -4X
please break down answer in long division form radther then synthetic division
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! here's how i did it and the answer i got.
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problem:
(2x^3 - 5x^2 + 22x + 51) divided by (2x + 3)
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first part of quotient is x^2 because 2x * x^2 = 2x^3
x^2 * (2x+3) = (2x^3 + 3x^2)
first remainder is (2x^3 - 5x^2 + 22x + 51) minus (2x^3 + 3x^2) = (-8x^2 + 22x + 51)
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second part of quotient is -4x because -4x * 2x = -8x^2
-4x * (2x+3) = (-8x^2 - 12x)
second remainder is (-8x^2 + 22x + 51) minus (-8x^2 - 12x) = (34x + 51)
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third part of quotient is 17 because 17 * 2x = 34x
17 * (2x+3) = (34x + 51)
third remainder is (34x + 51) minus (34x + 51) = 0
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the answer is:
(x^2 - 4x + 17)
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to prove, multiply (2x+3) * (x^2-4x+17) and you should get back to the original equation.
multiplying out, we get:
2x^3 -8x^2 + 34x + 3x^2 - 12x + 51
combining like terms, we get:
2x^3 -5x^2 + 22x + 51
this is the original equation so we're good.
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your question:
how did they get the -4x?
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1. you always divide the highest order of the divisor into the highest order of the dividend.
2. you then multiply the result of that division by the whole divisor.
3. you then subtract the result of that multiplication from the dividend to get the remainder which becomes the new dividend
4. you continue the first 3 steps until the highest order of the dividend is less than the highest order of the divisor. take that over the divisor and you have your final remainder.
in this problem, your final remainder was 0 / (2x+3) which became 0 which meant you had no remainder.
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the highest order of the divisor was 2x
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the first time the dividend was the original equation so we divided 2x into the highest order of the dividend which was x^3 to get x^2
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the second time the dividend was the first remainder so we divided 2x into the highest order of the dividend which was -8x^2 to get -4x.
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the third time the dividend was the second remainder so we divided 2x into the highest order of the dividend which was 34x to get 17.
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hope that helps.
let me know if you're still unsure how they got the -4x.
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