Chapter EP, Problem 6.8.11EP

To determine

To calculate: The zeros of the function $f\left(x\right)=3x^4-9x^2-12$ .

Answer to Problem 6.8.11EP

The zeros of the function $f\left(x\right)=3x^4-9x^2-12$ are $2,-2,i,-i$ .

Explanation of Solution

Given information:

The function $f\left(x\right)=3x^4-9x^2-12$ .

Formula used:

A polynomial of n degree has n zeros, which can be either real or imaginary.

Descartes’ rule of signs states that consider a polynomial $P\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ with real coefficients then, the number of times the sign between coefficients changes is equal to count of positive real zeroes of $P\left(x\right)$ or is less than this by an even number, the number of times the sign between coefficients changes is equal to count of negative real zeroes of $P\left(-x\right)$ or is less than this by an even number.

Calculation:

Consider the function $f\left(x\right)=3x^4-9x^2-12$ .

Observe that degree of polynomial is 4, so the functions has 4 zeros which can be either real or imaginary.

Descartes’ rule of signs states that consider a polynomial $P\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ with real coefficients then, the number of times the sign between coefficients changes is equal to count of positive real zeroes of $P\left(x\right)$ or is less than this by an even number.

So, count the number of times the sign changes between the coefficients of $f\left(x\right)=3x^4-9x^2-12$

There is 1 positive real zero.

Now,

  $f\left(-x\right)=3x^4-9x^2-12$

Descartes’ rule of signs states that consider a polynomial $P\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ with real coefficients then, the number of times the sign between coefficients changes is equal to count of negative real zeroes of $P\left(-x\right)$ or is less than this by an even number.

So, count the number of times the sign changes between the coefficients of $f\left(-x\right)=3x^4-9x^2-12$ .

There is 1 negative real zero.

Next, construct a table with possible combinations of real and imaginary zeros.

  $\begin{array}{cccc}\begin{array}{c}\text{Number of Positive}\\ \text{Real zeros}\end{array}& \begin{array}{c}\text{Number of Negative}\\ \text{Real zeros}\end{array}& \begin{array}{c}\text{Number of}\\ \text{Imaginary zeros}\end{array}& \begin{array}{c}\text{Total Number}\\ \text{of zeros}\end{array}\\ 1& 1& 2& 4\end{array}$

Recall that the Rational zero theorem states that provided a polynomial $P\left(x\right)$ with integral coefficients then every rational zero is of the form $\frac{p}{q}$ that is a rational number in simplest form. Here, p is a factor of the constant term and q is a factor of leading coefficient.

For the provided function leading coefficient is 3 and constant term is 12. Therefore, p is a factor of 12 and q is a factor of 3.

The possible combinations of $\frac{p}{q}$ in simplest form are,

  $\frac{p}{q}=\pm 1,\pm 2,\pm 3,\pm 4,\pm 12$

Next, construct a table with help of synthetic substitution to compute the value of $f\left(x\right)$ for real values of $x$ .

  $\begin{array}{cccccc}\frac{p}{q}& 3& 0& -9& 0& -12\\ -1& 3& -3& -6& 6& -18\\ 1& 3& 3& -6& -6& -18\\ 2& 3& 6& 3& 6& 0\\ -2& 3& -6& 3& -6& 0\end{array}$

As observed one zero is resulted at $x=2$ . Since, only one real positive zero is obtained so use the depressed polynomial to obtain remaining zeroes.

Now, the depressed polynomial is obtained is $f\left(x\right)=3x^3+6x^2+3x+6$ .

Group the terms together and factor the polynomial.

  $\begin{array}{c}3x^3+6x^2+3x+6=0\\ 3x^2\left(x+2\right)+3\left(x+2\right)=0\\ \left(x+2\right)\left(3x^2+3\right)=0\end{array}$

Apply the zero product property.

Either $\left(x+2\right)=0\text{ or }\left(3x^2+3\right)=0$

That is, $x=-2,x=\pm i$ .

Thus, the zeros of the function $f\left(x\right)=3x^4-9x^2-12$ are $2,-2,i,-i$ .