Chapter 4.5, Problem 22E

To determine

To find: a solution which is not a possible zero of the given function.

Answer to Problem 22E

The correct option is A.

Explanation of Solution

Given information:

Given function,

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

Given options are,

A.$-\frac{2}{3}$

B.$-\frac{3}{8}$

C.$\frac{3}{4}$

D.$\frac{4}{5}$

Calculation:

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

The rational root theorem states that “if $f\left(x\right)=a_n{x}^n+\mathrm{...}+a_{1}x+a_0$ has integer coefficient, then every rational solution of $f\left(x\right)=0$ has the form.

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

The factor of constant term $-36$ are $1,2,3,4,9,6,12,36$ and of leading coefficient 40 are $\pm 1,\pm 2,\pm 4,\pm 5,\pm 8,\pm 10,\pm 20,\pm 40$ .

Because 3 is not factor of $40$ , so $-\frac{2}{3}$ is not possible solution.

Therefore, the correct option is A.