How to factorize the following expression: $2x^4 + 9x^3+8x^2 +9x+2=0$

Question

How to factorize the following expression: $2x^4 + 9x^3+8x^2 +9x+2=0$

I have tried finding a root through putting values, but I can't find any factor.

Please help.

---

PS: I'm an elementary student, the question standard might not match with that of the community but any help would be sincerely appreciated.

Answer 1

Hint:

This is a special polynomial, as the coefficients are symmetric.

Now set

$$t:=x+\frac1x$$ so that $$t^2=x^2+\frac1{x^2}+2.$$

Express the equation in terms of $t$ and $t^2$.

$$2x^2+9x+8+\frac 9x+\frac2{x^2}=2t^2+9t+4=(2t+1)(t+4)=0.$$ $$2x^2+x+2=0,\\x^2+4x+1=0.$$

Answer 2

You probably have information about $4$ degree symmetric polynomials in your textbook. Since it is in my textbook, I thought of dividing directly each side $x ^ 2.$

Here is an alternative way:

$$2x^4 + 9x^3+8x^2 +9x+2=(x^2+ax+b)(2x^2+cx+d)$$

$$2x^4 + 9x^3+8x^2 +9x+2= 2x^4+x^3(c+2a)+x^2(2b+ac+d)+x(ad+bc)+bd$$

So, you get

$$\begin{cases} c+2a=9 \\ 2b+ac+d=8 \\ad+bc=9 \\ bd=2\end{cases}$$

For the next trick you can take $b=±1, d±2$ or $b±2, d=±1.$

Of course only $1$ pair will work.

Then you will get the quadratic equation respect to $a$ (or $c$).

You don't need to think for 1000 years :)