<s>[INST] Finding roots of the fourth degree polynomial: $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian
 [/INST] Guessing one root sometimes opens up the whole equation for you.
First notice that $\displaystyle x=1$ gives 0. So $\displaystyle x-1$ is a factor.
Next, rewrite as
$\displaystyle 2x^4 - 2x^3 + 5x^3 - 5x^2 -6x^2 + 6x - 15x + 15$
This is to try and get $x-1$ as a factor.
This gives us
$\displaystyle 2x^3(x-1) + 5x^2(x-1) - 6x(x-1) - 15(x-1) = (x-1)(2x^3 + 5x^2 - 6x - 15)$
Now notice that $\displaystyle 2x^3 - 6x = 2x(x^2-3)$ and $5x^2 - 15 = 5(x^2 - 3)$
Thus
$\displaystyle (x-1)(2x^3 + 5x^2 - 6x - 15) = (x-1)(2x(x^2 - 3) + 5(x^2 - 3)) = (x-1)(x^2-3)(2x+5)$
and so the roots are $\displaystyle 1, \pm\sqrt{3}, -\frac{5}{2}$
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