Chapter 3, Polynomial and Rational Functions - Section 3.4 - Real Zeros of Polynomials - 3.4 Exercises - Page 320: 78

$3$ is an upper bound for the real zeros of $P$ $-3$ is a lower bound for the real zeros of $P$

Upper and Lower bound Theorem states that: Suppose $f$ is a polynomial of degree $n\geq 1$. If $c>0$ is synthetically divided into $f$ and all of the coefficients of the quotient and remainder are all non-negative values, then $c$ is an upper bound for the real zeros of $f$. That is, there are no real zeros greater than $c$. If $c<0$ is synthetically divided into $f$ and the coefficients of the quotient and remainder alternate signs, then $c$ is a lower bound for the real zeros of $f$. That is, there are no real zeros less than $c$. If the number $0$ occurs as a coefficient of the quotient or remainder in the final line of the division table in either of the above cases, it can be treated as (+) or (−) as needed. $P(x)=2x^3-3x^2-8x+12$; let's try $3$ $\begin{array}{lllll} \underline{3}| & 2 & -3 & -8 & 12& \\ & & 6& 9 & 3 \\ \hline & & & & \\ & 2 & 3 & 1 & 15 \end{array}$ All of the coefficients of the quotient and remainder are all non-negative values. thus, $3$ is an upper bound for the real zeros of $P$. Let's try $-3$ $\begin{array}{lllll} \underline{-3}| & 2& -3 & -8 & 12 & \\ & & -6&27 & -57\\ \hline & & & & \\ & 2& -9& 19 & -45 \end{array}$ The coefficients of the quotient and remainder alternate signs, thus, $-3$ is a lower bound for the real zeros of $P$.