Long equation problems

Question

I am given this equation:

$$15x^6+31x^5-254x^4-352x^3+329x^2+9x-18=0$$

I know how to solve this question but the problem is it is too long to plug in numbers, what am I advised to do?

Answer 1

An easy thing to check is if it has rational roots. You only need check for factors of the form $ax+b$ where $a$ is a factor of $15$ and $b$ is a factor (positive or negative) of $18$.

You should for example find that $5x-3$ is a factor i.e $\frac {3}{5}$ is a root.

There's also a much simpler factor for you to find.

Divide through by these factors to obtain a quartic $$3x^4+2x^3-50x^2+2x+3 $$ which has no more rational roots.

Answer 2

$$f(x)=15x^6+31x^5-254x^4-352x^3+329x^2+9x-18$$ $$f(x)=(x+2) (5 x-3) \left(x^2-4 x+1\right) \left(3 x^2+14 x+3\right)$$ after factoring the quartic given by @S. Dolan