Chapter 5.8, Problem 3AGP

To determine

To calculate: The simplified form of the equation $h\left(x\right)=9x^4+5x^2-4$ ,

And find all the possible rational roots.

Answer to Problem 3AGP

The simplified form of the equation $h\left(x\right)=9x^4+5x^2-4$ has no rational roots.

Explanation of Solution

Given information:

The given equation:

  $h\left(x\right)=9x^4+5x^2-4$

Calculation:

Consider the expression as:

  $h\left(x\right)=9x^4+5x^2-4$

Rewrite the expression as:

  $\begin{array}{l}9x^4+5x^2-4\\ 3^2\cdot x^4+5x^2-4\end{array}$

As with the help of middle term factorization method solve the expression:

  $\begin{array}{c}9x^4+5x^2-4=0\\ 9x^4+9x^2-4x^2-4=0\end{array}$

Now factor out he greatest common factor:

  $\begin{array}{c}9x^4+9x^2-4x^2-4=0\\ 9x^2\left(x^2+1\right)-4\left(x^2+1\right)=0\\ \left(9x^2-4\right)\left(x^2+1\right)=0\end{array}$

Here these are the factors $\left(9x^2-4\right)\left(x^2+1\right)$ of the equation $9x^4+5x^2-4$ .

Now factor the term:

  $\left(9x^2-4\right)$

Here apply the formula of $a^2-b^2=\left(a+b\right)\left(a-b\right)$ .

So can write this as:

  $\begin{array}{l}9x^2-4\\ {\left(3x\right)}^2-{\left(2\right)}^2\\ \left(3x+2\right)\left(3x-2\right)\end{array}$

Now factor the term $\left(x^2+1\right)$ .

  $x^2+1$

Now take $f\left(x\right)=0$ ,

Here,

  $\begin{array}{c}f\left(x\right)=0\\ x^2+1=0\end{array}$

The leading coefficient is $1$ and the trilling coefficient is $1$ .

Now to find the list of positive rational solutions for the expression look at the factors of the constants.

That are:

  $-1,1$

Here the last two terms subtracting values from the cube of the first and want to get zero,

So as have to take the positive number, and mostly take the less than mid-way.

So take a startup with $1$ and check the result, then again tried to get zero.

  $\begin{array}{c}{x}^2+1=0\\ 1^2+1=0\\ 2=0\end{array}$

So the value is not zero its too grater than zero.

So check with the value $-1$ .

  $\begin{array}{c}{x}^2+1=0\\ -1^2+1=0\\ 2=0\end{array}$

Here no polynomial roots are found.

Thus, the simplified form of the equation $h\left(x\right)=9x^4+5x^2-4$ has no rational roots.