SOLUTION: Find the quotient and the remainder. Choose from below. (x^4-2x^2+3) divided by (x+1) A) x^3+x^2-x-1,2 B) x^3-x^2-x+1,2 C) x^2-3x+3,0 D) x^2-3x+4,-1

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find the quotient and the remainder. Choose from below. (x^4-2x^2+3) divided by (x+1) A) x^3+x^2-x-1,2 B) x^3-x^2-x+1,2 C) x^2-3x+3,0 D) x^2-3x+4,-1      Log On


   

Question 199762: Find the quotient and the remainder. Choose from below.
(x^4-2x^2+3) divided by (x+1)
A) x^3+x^2-x-1,2
B) x^3-x^2-x+1,2
C) x^2-3x+3,0
D) x^2-3x+4,-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%5E4+-+2x%5E2+%2B+3%29%2F%28x%2B1%29

First lets find our test zero:

x%2B1=0 Set the denominator x%2B1 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 -2x%5E2 there is a zero coefficient for x%5E3. This is simply because x%5E4+-+2x%5E2+%2B+3 really looks like 1x%5E4%2B0x%5E3%2B-2x%5E2%2B0x%5E1%2B3x%5E0
-1|10-203
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
-1|10-203
|
1

Multiply -1 by 1 and place the product (which is -1) right underneath the second coefficient (which is 0)
-1|10-203
|-1
1

Add -1 and 0 to get -1. Place the sum right underneath -1.
-1|10-203
|-1
1-1

Multiply -1 by -1 and place the product (which is 1) right underneath the third coefficient (which is -2)
-1|10-203
|-11
1-1

Add 1 and -2 to get -1. Place the sum right underneath 1.
-1|10-203
|-11
1-1-1

Multiply -1 by -1 and place the product (which is 1) right underneath the fourth coefficient (which is 0)
-1|10-203
|-111
1-1-1

Add 1 and 0 to get 1. Place the sum right underneath 1.
-1|10-203
|-111
1-1-11

Multiply -1 by 1 and place the product (which is -1) right underneath the fifth coefficient (which is 3)
-1|10-203
|-111-1
1-1-11

Add -1 and 3 to get 2. Place the sum right underneath -1.
-1|10-203
|-111-1
1-1-112

Since the last column adds to 2, we have a remainder of 2. This means x%2B1 is not a factor of x%5E4+-+2x%5E2+%2B+3
Now lets look at the bottom row of coefficients:

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

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

and the last coefficient 2, is the remainder, which is placed over x%2B1 like this

2%2F%28x%2B1%29



Putting this altogether, we get:

x%5E3+-+x%5E2+-+x+%2B+1%2B2%2F%28x%2B1%29

So %28x%5E4+-+2x%5E2+%2B+3%29%2F%28x%2B1%29=x%5E3+-+x%5E2+-+x+%2B+1%2B2%2F%28x%2B1%29

which looks like this in remainder form:
%28x%5E4+-+2x%5E2+%2B+3%29%2F%28x%2B1%29=x%5E3+-+x%5E2+-+x+%2B+1 remainder 2


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



If you have any questions, email me at jim_thompson5910@hotmail.com.
Check out my website if you are interested in tutoring.