SOLUTION: Graph the polynomial function P(x) = x^4 + x^3 -3x^2- 5x -2 to approximately find the function's zeros, then use synthetic division and the remainder theorem to exactly find its ze

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Question 90246: Graph the polynomial function P(x) = x^4 + x^3 -3x^2- 5x -2 to approximately find the function's zeros, then use synthetic division and the remainder theorem to exactly find its zeros.
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
When we graph, we find a zero at x=-1

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.
-1|11-3-5-2
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
-1|11-3-5-2
|
1

Multiply -1 by 1 and place the product (which is -1) right underneath the second coefficient (which is 1)
-1|11-3-5-2
|-1
1

Add -1 and 1 to get 0. Place the sum right underneath -1.
-1|11-3-5-2
|-1
10

Multiply -1 by 0 and place the product (which is 0) right underneath the third coefficient (which is -3)
-1|11-3-5-2
|-10
10

Add 0 and -3 to get -3. Place the sum right underneath 0.
-1|11-3-5-2
|-10
10-3

Multiply -1 by -3 and place the product (which is 3) right underneath the fourth coefficient (which is -5)
-1|11-3-5-2
|-103
10-3

Add 3 and -5 to get -2. Place the sum right underneath 3.
-1|11-3-5-2
|-103
10-3-2

Multiply -1 by -2 and place the product (which is 2) right underneath the fifth coefficient (which is -2)
-1|11-3-5-2
|-1032
10-3-2

Add 2 and -2 to get 0. Place the sum right underneath 2.
-1|11-3-5-2
|-1032
10-3-20

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 4 coefficients (1,0,-3,-2) form the quotient




So


-------------------------------------------------------------------
Now use the same test zero (which is -1) and perform synthetic division on


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 to there is a zero coefficient for . This is simply because really looks like
-1|10-3-2
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
-1|10-3-2
|
1

Multiply -1 by 1 and place the product (which is -1) right underneath the second coefficient (which is 0)
-1|10-3-2
|-1
1

Add -1 and 0 to get -1. Place the sum right underneath -1.
-1|10-3-2
|-1
1-1

Multiply -1 by -1 and place the product (which is 1) right underneath the third coefficient (which is -3)
-1|10-3-2
|-11
1-1

Add 1 and -3 to get -2. Place the sum right underneath 1.
-1|10-3-2
|-11
1-1-2

Multiply -1 by -2 and place the product (which is 2) right underneath the fourth coefficient (which is -2)
-1|10-3-2
|-112
1-1-2

Add 2 and -2 to get 0. Place the sum right underneath 2.
-1|10-3-2
|-112
1-1-20

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,-1,-2) form the quotient




So

Let's use the quadratic formula to solve for x:


Starting with the general quadratic



the general solution using the quadratic equation is:



So lets solve ( notice , , and )

Plug in a=1, b=-1, and c=-2



Negate -1 to get 1



Square -1 to get 1 (note: remember when you square -1, you must square the negative as well. This is because .)



Multiply to get



Combine like terms in the radicand (everything under the square root)



Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

or

Lets look at the first part:



Add the terms in the numerator
Divide

So one answer is




Now lets look at the second part:



Subtract the terms in the numerator
Divide

So another answer is


So our solutions are:
or

===================================

Answer:

So our zeros are:
(with a multiplicity of 3),
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