SOLUTION: Please help me with the following: Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x). P(x)=x^4+2x^3-2x+4 d(x)=x-2 Thank you.

Question 166602: Please help me with the following:
Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x).
P(x)=x^4+2x^3-2x+4
d(x)=x-2
Thank you.
Found 2 solutions by nerdybill, gonzo:
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x).
P(x)=x^4+2x^3-2x+4
d(x)=x-2
.
NOTE: it will be easier to see the solution (because of the spacing) below if you view the "source"!
.
x^3+5x^2+2x+2
_______________
x-2 |x^4+2x^3+0x^2-2x+4
x^4-3x^3
---------
5x^3+0x^2-2x+4
5x^3-2x^2
--------------
2x^2-2x+4
2x^2-4x
---------
2x+4
2x-4
----
8
.
Therefore:
Q(x) = x^3+5x^2+2x+2
R(x) = 8/(x^4+2x^3+0x^2-2x+4)
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
here's how it's done.
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you want to divide (x^4 + 2x^3 - 2x + 4) by (x-2).
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first you make sure your equation and your divisor are in the correct exponential order (highest exponent first, then next lower, then next lower, etc.)
they are that way already.
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next you divide the highest order of your divisor into the highest order of the equation to be divided which is the original equation.
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that would be (x^4) divided by (x) which equals (x^3).
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you multiply (x-2) * (x^3) to get (x^4 - 2x^3).
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you subtract that from your original equation.
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(x^4 + 2x^3 - 2x + 4) - (x^4 - 2x^3) is the same as
(x^4 + 2x^3 - 2x + 4)- (x^4) + (2x^3).
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your first remainder is: (4x^3 - 2x + 4).
the first term in your answer is (x^3).
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next you divide the highest order of your divisor into the highest order of the equation to be divided which would be your first remainder.
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that would be (4x^3) divided by (x) which equals (4x^2).
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you multiply (x-2) * (4x^2) to get (4x^3 - 8x^2)
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you subtract that from your first remainder.
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(4x^3 - 2x + 4) - (4x^3 - 8x^2) is the same as
(4x^3 - 2x + 4) - (4x^3) + (8x^2).
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your second remainder is (8x^2 - 2x + 4).
the second term in your answer is (4x^2).
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next you divide the highest order of your divisor into the highest order of the equation to be divided which would be your second remainder.
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that would be (8x^2) divided by (x) which equals (8x).
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you multiply (x-2) * (8x) to get (8x^2 - 16x).
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you subtract that from your second remainder.
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(8x^2 - 2x + 4) - (8x^2 - 16x) is the same as
(8x^2 - 2x + 4) - (8x^2) + (16x).
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your third remainder is (14x + 4).
the third term in your answer is (8x).
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next you divide the highest order of your divisor into the highest order of the equation to be divided which would be your third remainder.
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that would be (14x) divided by (x) which equals (14).
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you multiply (x-2) * (14) to get (14x - 28).
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you subtract that from your third remainder.
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(14x + 4) - (14x - 28) is the same as
(14x + 4) - (14x) + (28).
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your fourth remainder is (32).
the fourth term in your answer is (14).
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since the highest order of your equation to be divided is less then the highest order of the equation to be divided, you are done.
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your fourth remainder is your final remainder making it the remainder of the division.
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you add up all the terms in your answer to get:
(x^3 + 4x^2 + 8x + 14) + 32.
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in order to prove your answer is correct, you multiply this answer by (x-2) and you should get back to your original equation after adding in the remainder.
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(x-2) * (x^3 + 4x^2 + 8x + 14) are the factors to be multiplied.
first you multiply all terms of the equation by (x).
you get:
(x^4 + 4x^3 + 8x^2 + 14x) equals first part of multiplication.
then you multiply all terms of the equation by (-2).
you get:
(-2x^3 -8x^2 -16x -28) equals second part of multiplication.
you add the first part and the second part of your multiplication together to get.
(x^4 + 4x^3 - 2x^3 + 8x^2 - 8x^2 + 14x - 16x - 28) which becomes
(x^4 + 2x^3 - 2x - 28)
you add the remainder of 32 to this to make it:
(x^4 + 2x^3 - 2x + 4).
since this the original equation you started with, your division is good.
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