Solving a polynomial in an easier manner

I need to solve the following polynomial: $$(12x-1)(6x-1)(4x-1)(3x-1)=15$$

I tried do the multiplication and ended up with an even worse expression. There are a few other questions like this one on the list of exercises I'm trying to solve, and I just wanted an easier method, without having to do the "brutal" work.

Solution 1:

Hint: write it as: $$12\cdot 6\cdot 4 \cdot 3 \cdot \left(x-\frac{1}{12}\right)\left(x-\frac{1}{6}\right)\left(x-\frac{1}{4}\right)\left(x-\frac{1}{3}\right)=15$$

Note that the symmetric terms have equal sums $\left(x-\frac{1}{12}\right)+\left(x-\frac{1}{3}\right)=\left(x-\frac{1}{6}\right)+\left(x-\frac{1}{4}\right)$ $=2x - \frac{5}{12}$. This suggests the substitution $y=x-\frac{5}{24}\,$ which "shifts" the center of symmetry to $0\,$:

$$ 12\cdot 6\cdot 4 \cdot 3 \cdot \left(y+\frac{1}{8}\right)\left(y+\frac{1}{24}\right)\left(y-\frac{1}{24}\right)\left(y-\frac{1}{8}\right)=15 \\[5px] \iff \quad 12\cdot 6\cdot 4 \cdot 3 \cdot \left(y^2-\frac{1}{8^2}\right)\left(y^2-\frac{1}{24^2}\right) = 15 $$

The latter is a biquadratic which can be solved for $y^2\,$, which then gives $y\,$, then $x$.

Solution 2:

Expanding, we get

$$864 x^4 - 720 x^3 + 210 x^2 - 25 x - 14 = 0$$

Using the rational roots theorem, one can then verify that two roots are $x=-1/6$ and $x=7/12$. Factoring these out, we get

$$864 x^4 - 720 x^3 + 210 x^2 - 25 x - 14 = (6x+1)(12x-7)(12x^2-5x+2)$$

And of course the last quadratic is not factorable over $\mathbb R$.