Chapter 7.1, Problem 70STP

To determine

To find: The rational zeroes of the given function.

Answer to Problem 70STP

There are no rational zeroes of the given function.

Explanation of Solution

Given:

The given function is

  $f\left(x\right)=x^4-4x^3-7x^2+34x-24$

Calculation:

Calculate the rational zeroes of the function.

For a polynomial function with integer coefficients.

  $a_n{x}^n+a_{n-1}{x}^{n-1}\mathrm{....}{a}_0$

The rational roots can be found by checking for $\pm \frac{\text{dividers of }{a}_0}{\text{dividers of }{a}_n}$

Here

  $\begin{array}{l}{a}_0=24\\ a_n=1\end{array}$

The dividers of $a_0=1,2,3,4,6,8,12,24$

The dividers of $a_n=1$

The possible roots are:

  $\pm \frac{1,2,3,4,6,8,12,24}{1}$

Put these values in the given function and validate the roots. There are no solution found.

Therefore, there are no rational zeroes of the given function.