Factor

x4 - x3 - 7x2 + x + 6 = 0

Possible answers:

a. (x-6)
b. (x-3)
c. (x-2)
d. (x+6)

x4 - x3 - 7x2 + x + 6 = 0

We divide x4 - x3 - 7x2 + x + 6 
synthetically by each of those to find out which one gives a zero
remainder. I already know which one it is, so I'll try all the 
others first so you'll get the idea.

Let's try dividing x4 - x3 - 7x2 + x + 6
synthetically by (x-6). Change the sign of -6 to 6 for 
synthetic division:

  6 | 1  -1  -7   1   6
    |     6  30 138 834
      1   5  23 139 840

No,  840 is not 0, so

Let's try dividing x4 - x3 - 7x2 + x + 6
synthetically by (x-2). Change the sign of -2 to 2 for 
synthetic division:

  2 | 1  -1  -7   1   6
    |     2   2 -10 -18 
      1   1  -5  -9 -12

No, -12 is not 0, so

Let's try dividing x4 - x3 - 7x2 + x + 6
synthetically by (x+6). Change the sign of +2 to -6 for 
synthetic division:

 -6 | 1  -1  -7    1     6
    |    -6  42 -210  1254 
      1  -7  35 -209  1260

No, 1260 is not 0, so

Let's try dividing x4 - x3 - 7x2 + x + 6
synthetically by (x-3). Change the sign of -3 to 3 for 
synthetic division:

  3 | 1  -1  -7   1   6
    |     3   6  -3  -6 
      1   2  -1  -2   0

Yes!!!! That remainder is 0.  So the quotient is gotten from the
other numbers on the bottom, 1  2 -1 -2, which means 
1x3 + 2x2 - 1x - 2.  So we have factored x4 - x3 - 7x2 + x + 6 as

(x - 3)(x3 + 2x2 - x - 2) = 0

Now we can factor the x3 + 2x2 - x - 2 by grouping

x3 + 2x2 - x - 2

Factor x2 out of the first two terms and -1 out of the 
last two terms:

x2(x + 2) - 1(x + 2)

Then factor out (x + 2) and get

(x + 2)(x2 - 1)

Then factor the (x2 - 1) as (x - 1)(x + 1)

So the polynomial equation

x4 - x3 - 7x2 + x + 6 = 0

is now factored completely as

(x - 3)(x + 2)(x - 1)(x + 1) = 0

That's what you were told to do.  However they only gave you 
one of the factors so to get the problem right all you had to 
do was the first part and get (b).

Edwin
AnlytcPhil@aol.com