Determine all the factors for the following expression(note: you will need to use the rational root theorem more than one time) : f(x)=x^5+3x^4-5x^3-15x^2+4x+12?im so confused and this assignment is due tomorrow! ?

ANSWER

Possible rational roots: ±1,±2,±3,±4,±6,±12±1,±2,±3,±4,±6,±12

Actual rational roots: 1,−1,2,−2,−3

see attachments for all steps.

Possible roots are formed using the last term (12) of the given function and the 1st term (1) as your basis:

Since factors of 12 include plus or minus 1, 2, 3, 4, 6, 12,

possible rational roots include plus or minus 1/1, 2/1, 3/1, 4/1, 6,/1, 12/1 (and so on.)   I will take a chance and try the possible root 6/1, or just plain 6.

Use synthetic division, with 6 as the divisor and all of the coefficients of    x^5+3x^4-5x^3-15x^2+4x+12   as dividend:
       
  6  /  1   3   -5   -15   4    12
               6  54
       
          1   9   49   This is not going to work; 6 is not a root.  

Try -3 as divisor:

       
  -3  /  1   3   -5   -15   4    12
               -3    0     15  0   -12
       
          1    0   -5      0   4     0

Since the remainder is zero, x = -3 is a root of the given polynomial.  

Repeat this process, except skip x = -3 and x = 6 as divisors.

Use the coefficients    1   0   -5   0   4.  Note that plus or minus 4 over 1 forms other possible rational roots:  4/1, -4/1, 2/1, -2/1, 1, -1

Let's check out          
   
1 /  1  0  -5  0   4
           1  1  -4    -4

      1   1  -4  -4    0
Thus, 1 is also a root of the polynomial, along with -3.

Repeat this process.  As divisors try 4/1, -4/1, 2/1, -2/1, -1

Can you now finish this factoring?

The 2 roots found so far are {1, -3}.

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