Chapter 4.4, Problem 12E

To determine

To list: The possible rational roots of equation and then determine the rational roots.

Answer to Problem 12E

Possible rational roots: $\pm 1\text{, }\pm \text{2}$

Rational roots: 1, 2

Explanation of Solution

Given information: $x^4-5x^3+9x^2-7x+2=0$

Calculation:

  $x^4-5x^3+9x^2-7x+2=0$

Since leading coefficient is 1, rational root theorem says that possible rational roots are factors of free coefficient.

These are factors of coefficient $\left(-2\right):\pm 1\text{, }\pm \text{2}$

Pluck these numbers into equation. If the result is 0, than that number is rational root.

  $\begin{array}{l}x=1\\ 1^4-5\cdot 1^3+9\cdot 1^2-7\cdot 1+2=1-5+9-7+2=12-12=0\\ \to 1\text{ is rational root}\text{.}\end{array}$

  $\begin{array}{l}x=-1\\ {\left(-1\right)}^4-5\cdot {\left(-1\right)}^3+9\cdot {\left(-1\right)}^2-7\cdot \left(-1\right)+2=1+5+9+7+2=24\\ \to -1\text{ is not a rational root}\text{.}\end{array}$

  $\begin{array}{l}x=2\\ 2^4-5\cdot 2^3+9\cdot 2^2-7\cdot 2+2=16-40+36-14+2=54-54=0\\ \to 2\text{ is a rational root}\text{.}\end{array}$

  $\begin{array}{l}x=-2\\ {\left(-2\right)}^4-5\cdot {\left(-2\right)}^3+9\cdot {\left(-2\right)}^2-7\cdot \left(-2\right)+2=16+40+36+14+2=108\\ \to -2\text{ is not a rational root}\text{.}\end{array}$

Possible rational roots: $\pm 1\text{, }\pm \text{2}$

Rational roots: 1, 2