Question

In the complex number system, solve the equation$3x^5-10x^4+21x^3-42x^2+36x-8=0$

Short answer

The solution of the equation in the complex number system is$\left\{-2i,2i,1,2,\frac{1}{3}\right\}.$

Step-by-step solution

Step 1. Given Information

The given equation is $3x^5-10x^4+21x^3-42x^2+36x-8=0.$

We have to solve the equation.

Step 2. Solve

$$\begin{gathered} 3x^5-10x^4+21x^3-42x^2+36x-8=0 \\ \left(3x^5+21x^3+36x\right)-\left(10x^4+42x^2+8\right)=0 \\ 3x\left(x^4+7x^2+12\right)-2\left(5x^4+21x^2+4\right)=0 \\ 3x\left(x^4+4x^2+3x^2+12\right)-2\left(5x^4+20x^2+x^2+4\right)=0 \\ 3x\left[x^2\left(x^2+4\right)+3\left(x^2+4\right)\right]-2\left[5x^2\left(x^2+4\right)+1\left(x^2+4\right)\right]=0 \end{gathered}$$

Step 3. Solve

By proceeding with the calculation further

$$\begin{gathered} 3x\left(x^2+3\right)\left(x^2+4\right)-2\left(5x^2+1\right)\left(x^2+4\right) \\ \left(x^2+4\right)\left[3x\left(x^2+3\right)-2\left(5x^2+1\right)\right]=0 \\ \left(x^2+4\right)\left(3x^3+9x-10x^2-2\right)=0 \\ \left(x^2+4\right)\left(3x^3-3x^2-7x^2+9x-2\right)=0 \\ \left(x^2+4\right)\left[3x^2\left(x-1\right)-\left(7x^2-9x+2\right)\right]=0 \\ \left(x^2+4\right)\left[3x^2\left(x-1\right)-\left(7x^2-7x-2x+2\right)\right]=0 \end{gathered}$$

Step 4. Solve 

By proceeding with the calculation further

$$\begin{gathered} \left(x^2+4\right)\left[3x^2\left(x-1\right)-\left(x-1\right)\left(7x-2\right)\right]=0 \\ \left(x^2+4\right)\left(x-1\right)\left(3x^2-\left(7x-2\right)\right)=0 \\ \left(x^2+4\right)\left(x-1\right)\left(3x^2-7x+2\right)=0 \\ \left(x^2+4\right)\left(x-1\right)\left(3x^2-6x-x+2\right)=0 \\ \left(x^2+4\right)\left(x-1\right)\left(x-2\right)\left(3x-1\right)=0 \end{gathered}$$

Step 5. Writing the equation in the complex number system

As we know $\left(x^2+a^2\right)=\left(x+ai\right)\left(x-ai\right)\mathrm{and}{i}^2=-1.$

Therefore,

$\left(x+2i\right)\left(x-2i\right)\left(x-2\right)\left(3x-1\right)=0$

Thus, the solution of the equation in the number system is$\left\{-2i,2i,1,2,\frac{1}{3}\right\}.$