SOLUTION: Use synthetic division to determine if x - c is a factor of the given polynomial. 3x^4 + x^3 - 3x + 1; [x + (1/3)]

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Question 1208426: Use synthetic division to determine if
x - c is a factor of the given polynomial.
3x^4 + x^3 - 3x + 1; [x + (1/3)]



Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

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
Answer by math_tutor2020(3817)   (Show Source): 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.
-1/3310-31


Then we'll pull down the first coefficient (3)
-1/3310-31
3

Multiply that with the test root (-1/3).
(-1/3)*3 = -1
Place the result under the next coefficient.

-1/3310-31
-1
3


Then add straight down: 1 + (-1) = 0
-1/3310-31
-1
30


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
-1/3310-31
-10
300


Repeat the process again.
(-1/3)*0 = 0
-3+0 = -3
-1/3310-31
-100
300-3


Then one last time.
(-1/3)*(-3) = 1
1+1 = 2
-1/3310-31
-1001
300-32


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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