Question

use a graphing calculator and a system of equations to find the roots of the equation.
$x^{4}-4x^{3}=6x^{2}-12x$
what are the approximate values of the non - integral roots of the polynomial equation? choose two correct answers.
-1.95
0.21
4.73
-5.57
1.27

Explanation:

Step1: Rewrite the equation

Rewrite \(x^{4}-4x^{3}=6x^{2}-12x\) as \(x^{4}-4x^{3}-6x^{2}+12x = 0\). Then factor out \(x\): \(x(x^{3}-4x^{2}-6x + 12)=0\). By the rational - root theorem, we can test some simple values. We find that \(x = 0\) is a root.

Step2: Use a graphing calculator

Let \(y_1=x^{4}-4x^{3}\) and \(y_2 = 6x^{2}-12x\). When we graph \(y_1\) and \(y_2\) using a graphing calculator (or a graphing utility), we look for the \(x\) - values where \(y_1=y_2\).
The roots of the equation \(x^{4}-4x^{3}-6x^{2}+12x=0\) can be found by graphing. We know that \(x = 0\) is a root. For the non - zero roots, we can use the graphing method.
The roots of the polynomial equation \(x^{4}-4x^{3}-6x^{2}+12x = 0\) (solved by graphing \(y=x^{4}-4x^{3}-6x^{2}+12x\) or using a system \(y_1=x^{4}-4x^{3}\) and \(y_2=6x^{2}-12x\)):
We can also factor the cubic part \(x^{3}-4x^{2}-6x + 12\) further. By graphing, we find the non - integral roots.

Answer:

\(-1.95\), \(4.73\)