Question 1208426
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Think of 3x^4+x^3-3x+1 as 3x^4+1x^3+0x^2-3x+1
The coefficients from left to right are 3,1,0,-3,1
Write those coefficients along the top row of the synthetic division table. To the left of these coefficients is the test root -1/3. This is from solving x + 1/3 = 0 to get x = -1/3.


This is the set up.
<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>


Then we'll pull down the first coefficient (3)
<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>3</td><td></td><td></td><td></td><td></td></tr></table>
Multiply that with the test root (-1/3). 
(-1/3)*3 = -1
Place the result under the next coefficient.


<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>3</td><td></td><td></td><td></td><td></td></tr></table>


Then add straight down: 1 + (-1) = 0
<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>3</td><td>0</td><td></td><td></td><td></td></tr></table>


We'll repeat this process of multiplying with the test root and adding straight down to get this next column of values.
(-1/3)*0 = 0
0+0 = 0
<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td>-1</td><td>0</td><td></td><td></td></tr><tr><td></td><td></td><td>3</td><td>0</td><td>0</td><td></td><td></td></tr></table>


Repeat the process again.
(-1/3)*0 = 0
-3+0 = -3
<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td>-1</td><td>0</td><td>0</td><td></td></tr><tr><td></td><td></td><td>3</td><td>0</td><td>0</td><td>-3</td><td></td></tr></table>


Then one last time.
(-1/3)*(-3) = 1
1+1 = <font color=red>2</font>
<table border = "1" cellpadding = "5"><tr><td>-1/3</td><td></td><td>3</td><td>1</td><td>0</td><td>-3</td><td>1</td></tr><tr><td></td><td></td><td></td><td>-1</td><td>0</td><td>0</td><td>1</td></tr><tr><td></td><td></td><td>3</td><td>0</td><td>0</td><td>-3</td><td><font color=red size=4>2</font></td></tr></table>


The value in the bottom right corner is the remainder. I have highlighted it in <font color=red>red</font>


The nonzero remainder means that x+(1/3) is not a factor of 3x^4 + x^3 - 3x + 1


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Here's another method that doesn't involve synthetic division.


If x-c is a factor of f(x), then f(c) = 0. This is a special case of the remainder theorem.
Rewrite x + (1/3) as x - (-1/3) to determine c = -1/3.


We'll plug this into the polynomial to see if we get zero or not.
f(x) = 3x^4 + x^3 - 3x + 1
f(-1/3) = 3(-1/3)^4 + (-1/3)^3 - 3(-1/3) + 1
f(-1/3) = 3(1/81) - 1/27 + 1 + 1
f(-1/3) = 1/27 - 1/27 + 1 + 1
f(-1/3) = 2
The nonzero result tells us that x+(1/3) is not a factor of 3x^4 + x^3 - 3x + 1.
Note: The result 2 is the remainder we got in the previous section.



Yet another method involves graphing y = 3x^4 + x^3 - 3x + 1.
Notice how the curve doesn't pass through the x axis when x = -1/3.
This visually confirms our answer above.
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