SOLUTION: Find the quotient and remainder of f(x)=x^4-2 divided by p(x)=x-1.

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Question 92464This question is from textbook Algebra and Trigonometry
: Find the quotient and remainder of f(x)=x^4-2 divided by p(x)=x-1. This question is from textbook Algebra and Trigonometry

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Start with the given polynomial %28x%5E4+-+2%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 1x%5E4 to -2 there is a zero coefficient for x%5E3, x%5E2, and x . This is simply because x%5E4+-+2 really looks like 1x%5E4%2B0x%5E3%2B0x%5E2%2B0x%5E1%2B-2x%5E0
1|1000-2
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
1|1000-2
|
1

Multiply 1 by 1 and place the product (which is 1) right underneath the second coefficient (which is 0)
1|1000-2
|1
1

Add 1 and 0 to get 1. Place the sum right underneath 1.
1|1000-2
|1
11

Multiply 1 by 1 and place the product (which is 1) right underneath the third coefficient (which is 0)
1|1000-2
|11
11

Add 1 and 0 to get 1. Place the sum right underneath 1.
1|1000-2
|11
111

Multiply 1 by 1 and place the product (which is 1) right underneath the fourth coefficient (which is 0)
1|1000-2
|111
111

Add 1 and 0 to get 1. Place the sum right underneath 1.
1|1000-2
|111
1111

Multiply 1 by 1 and place the product (which is 1) right underneath the fifth coefficient (which is -2)
1|1000-2
|1111
1111

Add 1 and -2 to get -1. Place the sum right underneath 1.
1|1000-2
|1111
1111-1

Since the last column adds to -1, we have a remainder of -1. This means x-1 is not a factor of x%5E4+-+2
Now lets look at the bottom row of coefficients:

The first 4 coefficients (1,1,1,1) form the quotient

x%5E3+%2B+x%5E2+%2B+x+%2B+1

and the last coefficient -1, is the remainder, which is placed over x-1 like this

-1%2F%28x-1%29



Putting this altogether, we get:

x%5E3+%2B+x%5E2+%2B+x+%2B+1%2B-1%2F%28x-1%29

So %28x%5E4+-+2%29%2F%28x-1%29=x%5E3+%2B+x%5E2+%2B+x+%2B+1%2B-1%2F%28x-1%29

which looks like this in remainder form:
%28x%5E4+-+2%29%2F%28x-1%29=x%5E3+%2B+x%5E2+%2B+x+%2B+1 remainder -1


You can use this online polynomial division calculator to check your work