SOLUTION: What is the remainder when {{{x^7-x^6+x^5-x^4+4}}} is divided by {{{x-1}}}

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Question 149767: What is the remainder when x%5E7-x%5E6%2Bx%5E5-x%5E4%2B4 is divided by x-1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's simplify this expression using synthetic division


Start with the given expression %28x%5E7+-+x%5E6+%2B+x%5E5+-+x%5E4+%2B+4%29%2F%28x-1%29

First lets find our test zero:

x-1=0 Set the denominator x-1 equal to zero

x=1 Solve for x.

so our test zero is 1


Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from -x%5E4 to 4 there is a zero coefficient for x%5E3, x%5E2, and x. This is simply because x%5E7+-+x%5E6+%2B+x%5E5+-+x%5E4+%2B+4 really looks like 1x%5E7%2B-1x%5E6%2B1x%5E5%2B-1x%5E4%2B0x%5E3%2B0x%5E2%2B0x%5E1%2B4x%5E0
1|1-11-10004
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
1|1-11-10004
|
1

Multiply 1 by 1 and place the product (which is 1) right underneath the second coefficient (which is -1)
1|1-11-10004
|1
1

Add 1 and -1 to get 0. Place the sum right underneath 1.
1|1-11-10004
|1
10

Multiply 1 by 0 and place the product (which is 0) right underneath the third coefficient (which is 1)
1|1-11-10004
|10
10

Add 0 and 1 to get 1. Place the sum right underneath 0.
1|1-11-10004
|10
101

Multiply 1 by 1 and place the product (which is 1) right underneath the fourth coefficient (which is -1)
1|1-11-10004
|101
101

Add 1 and -1 to get 0. Place the sum right underneath 1.
1|1-11-10004
|101
1010

Multiply 1 by 0 and place the product (which is 0) right underneath the fifth coefficient (which is 0)
1|1-11-10004
|1010
1010

Add 0 and 0 to get 0. Place the sum right underneath 0.
1|1-11-10004
|1010
10100

Multiply 1 by 0 and place the product (which is 0) right underneath the sixth coefficient (which is 0)
1|1-11-10004
|10100
10100

Add 0 and 0 to get 0. Place the sum right underneath 0.
1|1-11-10004
|10100
101000

Multiply 1 by 0 and place the product (which is 0) right underneath the seventh coefficient (which is 0)
1|1-11-10004
|101000
101000

Add 0 and 0 to get 0. Place the sum right underneath 0.
1|1-11-10004
|101000
1010000

Multiply 1 by 0 and place the product (which is 0) right underneath the eighth coefficient (which is 4)
1|1-11-10004
|1010000
1010000

Add 0 and 4 to get 4. Place the sum right underneath 0.
1|1-11-10004
|1010000
10100004



Since the last column adds to 4, we have a remainder of 4.


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Answer:

So when x%5E7-x%5E6%2Bx%5E5-x%5E4%2B4 is divided by x-1, the remainder is 4