bounds test?

Question

Use the bounds test to determine the integer low and upper bounds for the real zeros of f(x)=x^5+5x^4-x^3+5x^2+4x-3

Answer 1

(Please ignore the dots in the divisions below; they are there for spacing purposes only)

Use synthetic division to show that -6 is a lower bound:

.....1 5 -1....5... 4....-3
-6....-6. 6 -30 150 -924
.....1 -1 5 -25 154 -927

The bottom row alternates, so -6 is a lower bound
(it happens to be the greatest integer upper bound)

1 is an upper bound:

.... 1 5 -1...5..4..-3
1......1..6...5.10 14
.....1 6..5 10.14 11

The bottom row is all positive, so 1 is an upper bound
(it happens to be the least upper integer bound)