Chapter 5.6, Problem 43HP

To determine

To explain: the zeros of a function can be located by using the Remainder theorem and making a table of values for different input values and then comparing the remainders.

Explanation of Solution

Given information:

The zeros of a function can be located by using the Remainder theorem and making a table of values for different input values and then comparing the remainders.

Formula used:

Location principle : Suppose $y=f(x)$ represents a polynomial function and a and b are two real numbers such that $f(a)<0\text{ and }f(b)>0.$ Then the function has at least one real zero between a and b .

Remainder Theorem:- If a polynomial $P(x)$ is divided by $x-r$ , the remainder s a constant $P(r)$ , and

  $P(x)=Q(x)(x-r)+P(r)$ ,

Where $Q(x)$ is a polynomial with degree one less than $P(x)$ .

Proof:

Let the function be $f(x)$ .

Let c be a zero of the function . This implies $f(c)=0$ .

Let a and b are two real numbers .

According to the location principle if ,

  $f(a)<0\text{ and }f(b)>0$ we get at least one real zero between a and b .

The location of zeros is to be done by Remainder Theorem.

Further, taking different values of x we get the value of the function.

This will provide us with the values of x where there is change in sign of the function.

Hence proved.