Chapter 3.4, Problem 71E

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 $P\left(x\right)=2x^3+5x^2+x-2$

  $a=-3$

  $b=1$

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 $x+3\text{ and }x-1$ using synthetic division.

1	2	5	1	-2
		2	7	8
	2	7	8	6

-3	2	5	1	-2
		-6	3	-12
	2	-1	4	-14

By the upper and lower bonds theorem, -3 is a lower bond and 1 is the upper bond for the zeros.