Chapter 3.4, Problem 73E

To determine

To show:that the value of $a\text{ and }b$ are lower and upper bounds for real zeros of the polynomial.

Explanation of Solution

Given information:

the polynomial is $P\left(x\right)=8x^3+10x^2-39x+9$

  $a=-3$

  $b=2$

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 $x+3\text{ and }x-2$ using synthetic division.

2	8	10	-39	9
		16	52	26
	8	26	13	35

-3	8	10	-39	9
		-24	42	-9
	8	-14	3	0

By the upper and lower bonds theorem, -3 is a lower bond and 2 is the upper bond for the zeros.