Answer
a. Upper bound $b=1$, see prove below.
b. Lower bound $a=-3$, see prove below.
Work Step by Step
a. Upper bound $b=1$, we use synthetic division to divide the polynomial by $x-1$ as shown in the figure.
As can be seen, all the numbers $2,7,8,6$ in the line containing the quotient and remainder of this division are positive. Based on the Upper and Lower Bound Theorem (page 279 in the book), we conclude that $b=1$ is an upper bound of the function $P(x)$.
b. Lower bound $a=-3$, similarly, we used synthetic division to divide $P(x)$ by $x+3$ as shown in the figure,
and it can be seen that the numbers of the quotient and remainder line have alternating signs $2,-1,4,-14$.
Based on the Upper and Lower Bound Theorem (page 279 in the book), we conclude that $a=-3$ is a lower
bound of the function $P(x)$.
