The number of irrational roots of $(x^2 -3x +1)(x^2 +3x +2)(x^2 -9x + 20) = -30$ is___?

Question

The number of irrational roots of $(x^2 -3x +1)(x^2 +3x +2)(x^2 -9x + 20) = -30$ is___?

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My Approach :

Multiplying the above equation and then apply Descartis rule is very lengthy method. Also I tried it by taking factors of $30$ and then solving the equation but that is also going very lengthy .

Please suggest some good method of doing this.

Answer 1

I'm not sure if this is a good method for your question, but we have $$(x^2-3x+1)(x+1)(x+2)(x-4)(x-5)=-30$$ rearranging $$(x^2-3x+1)(x+1)(x-4)(x+2)(x-5)=-30$$ $$(t+1)(t-4)(t-10)=-30$$ where $t=x^2-3x$.

Now, one sees that $t=5$ works. So, $$(t+1)(t-4)(t-10)+30$$ is divisible by $t-5$ to have $$(t+1)(t-4)(t-10)+30=(t-5)(t^2-8t-14)$$ So, $$x^2-3x=t=5,4\pm\sqrt{30}$$

Answer 2

Note that the polynomial is a sixth degree monic polynomial whose constant term is $70$. Hence, by rational roots theorem, the possible roots can only be divisors of $70$. You will find that none of the roots are rational.