If [x+(1/3)] or [x-(-1/3)] were a factor then the factorization would be
Then would give a zero of .
So to find out, we use synthetic division to see if the remainder, which is the
same value as when -1/3 is substituted for x, is zero. That happens when the
last term on the bottom right of the synthetic division is 0. So here goes the
synthetic division:
Be sure to remember that has a missing term in
. So we must consider it as
-1/3 | 3 1 0 -3 1
| -1 0 0 1
3 0 0 -3 2
The number on the bottom right did not turn out to be zero. It came out to be
2 instead, so, no [x + (1/3)] is not a factor of .
Edwin
You can put this solution on YOUR website!
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.
-1/3
3
1
0
-3
1
Then we'll pull down the first coefficient (3)
-1/3
3
1
0
-3
1
3
Multiply that with the test root (-1/3).
(-1/3)*3 = -1
Place the result under the next coefficient.
-1/3
3
1
0
-3
1
-1
3
Then add straight down: 1 + (-1) = 0
-1/3
3
1
0
-3
1
-1
3
0
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
-1/3
3
1
0
-3
1
-1
0
3
0
0
Repeat the process again.
(-1/3)*0 = 0
-3+0 = -3
-1/3
3
1
0
-3
1
-1
0
0
3
0
0
-3
Then one last time.
(-1/3)*(-3) = 1
1+1 = 2
-1/3
3
1
0
-3
1
-1
0
0
1
3
0
0
-3
2
The value in the bottom right corner is the remainder. I have highlighted it in red
The nonzero remainder means that x+(1/3) is not a factor of 3x^4 + x^3 - 3x + 1
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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