Solving with the RRT Solutions

• Use    technology    to:

  • view the graph,

  • determine the rational roots and name any integer roots with RRT, and

  • identify any multiplicities that exist for any root.

• Then, use one or more of these methods to find the actual roots algebraically:
 
  •    synthetic division    
  •    factoring   
 
  •    quadratic formula   

Example 6

Determine all roots of the function.

$f\left(x\right)=x^4+8x^3–10x^2–8x+9$

Note

The answer could also be written as the product of its factors.

$f\left(x\right)=(x+1){(x–1)}^2(x+9)$

 

There are many ways to solve these problems that depend on the order in which you use the possible rational roots.

Example 7

Determine all roots of the function. Then write the equation as a product of its factors using rational numbers.

$9x^3+4x=45x^2+20$

$\begin{array}{rcl}9x^2+4& =& 0\\ 9x^2& =& –4\\ x^2& =& –\frac{4}{9}\\ \sqrt{x^2}& =& \pm \sqrt{–\frac{4}{9}}\\ x& =& \pm \frac{2i}{3}\end{array}$

Example 8

Determine all roots of the function. Then write the function as a product of factors using rational numbers.

$g\left(x\right)=x^5–14x^3–16x^2+24x+32$

$$\begin{gathered} x^2–2=0 \\ {x}^2=2 \\ x=\pm \sqrt{2} \end{gathered}$$

$g\left(x\right)={\left(x+2\right)}^2\left(x–4\right)\left(x^2–2\right)$

Note

There are many ways to solve these problems that depend on the order in which you use the possible rational roots.

Example 9

Determine all roots of the function.

$h\left(x\right)=x^3–5x+4$

$\mathrm{RRT}=\frac{\pm 1,\pm 2, \pm 4}{\pm 1} =\pm 1, \pm 2, \pm 4$

$a=1, b=1, c=–4$