Chapter 3.4, Problem 76E

To determine

To find:the integers that are lower and upper bounds for real zeros of the polynomial.

Answer to Problem 76E

The lower and upper bounds for real zeros of the polynomial is $3\text{ and }-3$ .

Explanation of Solution

Given information:

the polynomial is $P\left(x\right)=2x^3-3x^2-8x+12$

concept used:

Let P be a polynomial with real coef?cients.

1. If we divide $P\left(x\right)$ by $x-b$ (with $b>0$ ) using synthetic division and if the row that contains the quotient and remainder has no negative entry, then b is an upper bound for the real zeros of P.

2. If we divide $P\left(x\right)$ by $x-a$ (with $a>0$ ) using synthetic division and if the row that contains the quotient and remainder has entries that are alternately nonpositive and nonnegative, then a is a lower bound for the real zeros of P.

Proof:

Divide the $P\left(x\right)$ by $x-3\text{ and }x+3$ using synthetic division.

3	2	-3	-8	12
		6	9	3
	2	3	1	15

-3	2	-3	-8	12
		-6	27	-57
	2	-9	19	-45

By the upper and lower bonds theorem, -3 is a lower bond and 3 is the upper bond for the zeros.

Since neither -3 or 3 is a zero(the remainder are not zero in the division table, so all the real zeros lie between these numbers.