Chapter 4.3, Problem 35E

To determine

To Find: Thenumber of times $2$ is the root of the $x^6-9x^4+24x^2-16=0$ .

Answer to Problem 35E

The number 2 is the factor of the given polynomial 2 times.

Explanation of Solution

Consider the given equation is,

  $x^6-9x^4+24x^2-16=0$

Consider the synthetic division for the given polynomial as,

  $\begin{array}{l}\overline{)2}10-90240-16\\ \underset{\_}{24-10-20816}\\ 12-5-10480\end{array}$

Since, the remainder is zero, therefore 2 is the root of the given equation.

Consider the depressed polynomial is,

  $x^5+2x^4-5x^3-10x^2+4x+8=0$

Consider the synthetic method to check that 2 is the root of the equation is,

  $\begin{array}{l}\overline{)2}12-5-1048\\ \underset{\_}{286-8-8}\\ 143-4-40\end{array}$

Consider the depressed polynomial is,

  $x^5+4x^3+3x^2-4x-4x=0$

Consider the synthetic method to check that 2 is the root of the equation is,

  $\begin{array}{l}\overline{)2}143-4-4\\ \underset{\_}{26910}\\ 16956\end{array}$

Thus, the number 2 is the factor of the given polynomial 2 times.