Question

Find the complex zeros of:

$f\left(x\right)=5x^4-9x^3-7x^2-31x-6$

Short answer

The two complex zeros of the function is:$x=\frac{-1\pm i\sqrt{7}}{2}$

Step-by-step solution

Step 1. Given information

The function is:

$f\left(x\right)=5x^4-9x^3-7x^2-31x-6$

Step 2.  Draw the graph of the function by using graphing calculator.

The graph of the function is:

From this graph it is clear that one of the real zero of the function is $x=3$

So$(x-3)$will be one factor of$f\left(x\right)$.

Step 3. To find the depressed equation.

Since $f\left(3\right)=0$

So, by using synthetic division:

localid="1647766430444" $35-9-7-31-61518336561120$

So, $f\left(x\right)=(x-3)(5x^3+6x^2+11x+2)$

So depresses equation is:

localid="1647764958561" $5x^3+6x^2+11x+2=0$

The one of the real root of this depressed equation is$$\begin{gathered} x=-\frac{1}{5} \\ x=-0.2 \end{gathered}$$

Step 4. To factorize depressed equation.

$-0.256112-1-1-255100$

The depressed equation can be factored as:

role="math" localid="1647766777287" $(x+0.2)(5x^2+5x+10)$

Step 5. Solve the quadratic equation.

$5x^2+5x+10=0$

$$\begin{gathered} x=\frac{-5\pm \sqrt{5^2-4.5.10}}{2.5} \\ x=\frac{-5\pm \sqrt{25-200}}{10} \\ x=\frac{-5\pm \sqrt{-175}}{10} \\ x=\frac{-5\pm \sqrt{175i^2}}{10} \\ x=\frac{-5\pm i5\sqrt{7}}{10} \\ x=\frac{5(-1\pm i\sqrt{7)}}{10} \\ x=\frac{-1\pm i\sqrt{7}}{2} \end{gathered}$$

So there are two complex zeros of the given function.$x=\frac{-1\pm i\sqrt{7}}{2}$