Given `x^7-7x^6+x^5-3x^4+x^2-2x+3=0` , determine if -1 is a lower bound on the roots, and if 2 is an upper bound on the roots.

We use synthtetic division and the following theorem: If after applying synthetic division for a prospective root the resulting coefficients alternate from nonnegative to nonpositive etc..., then the prospective root is a lower bound on the real roots. If the resulting coefficients are all nonnegative, the prospective root is an upper bound on the real roots.

(1) The coefficients for synthetic division are 1,-7,1,-3,0,1,-2,3.

(2) First we try -1:

-1| 1   -7   1   -3   0   1   -2   3
1  -8   9  -12  12  -11 9   -6

Notice the alternating signs, so -1 is a lower bound on the real roots.

(3) Next we try 2:

2| 1   -7   1   -3    0    1    -2     3
1   -5  -9   -21-42  -83  -168 -333

The coefficients are not all nonnegative, so 2 is not an upper bound on the real roots.