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
