Chapter 2, Problem 55RE

Solution

To find: All the real zeros of the function $f\left(x\right)=x^3-2x^2-8x+5$ and find exact values if possible. Identify each zeros rational or irrational or complex.

The real zeros are $x=-2.34,0.57,3.77$ .

Given information:

The given function is $f\left(x\right)=x^3-2x^2-8x+5$ .

Concept used:

Factor theorem: A polynomial function has a factor $x-k$ if only if $f\left(k\right)=0$ .

Rational zero theorem: Suppose f is a function of degree $n\ge 1$ of the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{......}+a_0$ with every coefficient an integer and $a_0\ne 0.$ If $x=\frac{p}{q}$ is a rational zero of $f$ where $p$ and $q$ have no common integers other than $\pm 1$ then $p$ is an integer factor of the constant $a_0$ and $q$ is an integer factor of the leading coefficient $a_0$ .

Calculations:

  $f\left(x\right)=x^3-2x^2-8x+5$

The possible rational zeros are $\pm 1,\pm 5$ .

Now check the zeros:

  $\begin{array}{c}f\left(1\right)={\left(1\right)}^3-2{\left(1\right)}^2-8\left(1\right)+5\\ =1-2-8+5\\ =-4\end{array}$

Thus, $x=-1$ is not rational zeros.

  $\begin{array}{c}f\left(-1\right)={\left(-1\right)}^3-2{\left(-1\right)}^2-8\left(-1\right)+5\\ =-1-2+8+5\\ =10\end{array}$

Thus, $x=1$ is not rational zero.

  $\begin{array}{c}f\left(-5\right)={\left(-5\right)}^3-2{\left(-5\right)}^2-8\left(-5\right)+5\\ =-125-50+40+5\\ =-130\end{array}$

Thus, $x=-5$ is not rational zero.

  $\begin{array}{c}f\left(5\right)={\left(5\right)}^3-2{\left(5\right)}^2-8\left(5\right)+5\\ =125-50-40+5\\ =40\end{array}$

Thus, $x=5$ is not rational zero.

This is prime equation and cannot factor it.

There is not any exact real zeros. Now find the rational zeros using graph.

The graph of function,

  

From the graph the real zeros are $x=-2.34,0.57,3.77$ .