x⁴- 3x³ - 21x² + 43x + 60
Possible zeros are ± the factors of 60:
±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60
It's a matter of trial and error:
try -1
-1| 1 -3 -21 43 60
| -1 4 17 -60
1 -4 -17 60 0
Yes 1 is a zero. So the factorization so far is
(x-1)(x³-4x²-17x+60)
The possible zeros of x³-4x²-17x+60 are still
±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60
try 1
1| 1 -4 -17 60
| 1 -3 -20
1 -3 -20 40
1 is not a zero of x³-4x²-17x+60
try -1
-1| 1 -4 -17 60
| -1 5 12
1 -5 -12 72
-1 is not a zero of x³-4x²-17x+60
try 2
2| 1 -4 -17 60
| 2 -4 -42
1 -2 -21 18
2 is not a zero of x³-4x²-17x+60
try -2
-2| 1 -4 -17 60
| -2 12 10
1 -6 -5 70
-2 is not a zero of x³-4x²-17x+60
try 3
3| 1 -4 -17 60
| 3 -3 -60
1 -1 -20 0
Yes 3 is a zero of x³-4x²-17x+60. So the factorization so far is
(x-1)(x-3)(x²-x-20)
We can factor the trinomial in the last parentheses without
using synthetix division:
(x-1)(x-3)(x-5)(x+4)
That's the final factorization:
The zeros are 1,3,5,-4.
Edwin
