Solve the equation, $$6x^5 + 5x^4 − 51x^3 + 51x^2 − 5x − 6 = 0$$(hints: the pattern of the coefficients)
How I attempted this problem is to use the rational root theorem to obtain the factors. The possible roots are $$±(1, 1/2, 1/3, 1/6, 2, 2/3, 3, 3/2, 6)$$ By substituting the values inside the equation, only the following roots satisfy the equation:
Henceforth, I came up with the following based on the above roots: $$\left(x-1\right)\left(2x-3\right)\left(3x-2\right)$$ Clearly expanding them would give me a 3rd degree polynomial as follows: $$6x^3 - 19x^2 + 19x - 6$$
Using polynomial division where I divided the original 5th degree equation with the above equation, I obtained the following equation: $$x^2+4x+1$$ Now, solving the above equation using quadratic formula, I am able to get the roots. Hence, I have obtained all the roots of the solution. However, looking at the hint that asks me to observe the pattern of the coefficients, I think that the method used to arrive at my solution might have been long-winded (it was indeed tedious plugging in the rational .roots one-by-one into the equation to see which returns value of 0). Is there something that I'm missing in the way I approached the question?