Chapter 2.6, Problem 40E

Solution

To find: what is the largest number of possible rational zeros of $f$ .

The number of possible rational zeros is $r^{°}S$ .

Given information: 

The function $f\left(x\right)=40x^5-42x^4-107x^3+107x^2+33x-36$ .

Formula used:

The Rational Zero Theorem:

If $f\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ consists of integer coefficients, then the shape of each rational zero of $f$ is as follows:

  $\frac{p}{q}=\frac{\text{ factor of constant term }{a}_0}{\text{ factor of leading coefficient }{a}_n}$

Calculation:

The divisors of the leading coefficient are:

  $p_1,p_2,p_3,\dots ,p_{\tau }$

The divisors of the constant term are:

  $q_1,q_2,q_3,\dots ,q_s$

It considered all the divisors (positive and negative).

The possible rational zeros of the function are in the form:

  $\frac{q_k}{p_j}$

Count the maximum number of possible fractions of this type. This number can be obtained when $a_n$ and $a_0$ have no common divisors. In this case all the fractions of this kind are different (they cannot be simplified):

  $\begin{array}{l}\frac{q_1}{p_1},\frac{q_1}{p_2},\dots ,\frac{q_1}{p_r}\\ \frac{q_2}{p_1},\frac{q_2}{p_2},\dots ,\frac{q_2}{p_r}\\ \frac{q_s}{p_1},\frac{q_s}{p_2},\dots ,\frac{q_s}{p_r}\end{array}$

Therefore, the number of possible rational zeros is $r^{°}S$ .