Question

How do you factor `y= 6x^3+13x^2-14x+3` ?

Answer 1

`color(red)((2x-1)(3x-1)(x+3))`

Explanation:

We can optimistically hope that `y=color(lime)6x^3+13x^2-14x+color(magenta)3`
has at least one rational root.

By the Rational Root Theorem we know that any such root must be of the form `color(blue)(p/q)`
where `color(blue)p` is an integer factor of `color(magenta)3` (for this expression)
`color(white)("XXX")color(blue)p in {+-1,+-3}`
and
`color(blue)q` is an integer factor of `color(lime)6` (again, for this expression)
`color(white)("XXX")color(blue)q in {+-1,+-2,+-3,+-6}`

There turn out to only be 10 possible rational roots using this information:
`color(white)("XXX")color(blue)(p/q) in {+-3,+-1,+-1/2,+-1/3,+-1/6}`

You could evaluate these manually, but I used a spreadsheet:

In this case, we have been extremely lucky, since we have all 3 possible roots identified `{0.5=1/2, 0.333bar3=1/3, -3}`

Which implies the factors:
`color(white)("XXX")color(lime)6 * (x-1/2) * (x-1/3) * (x+3)`
or (after using the `color(lime)6` to clear the fractions)
`color(white)("XXX")(2x-1)(3x-1)(x+3)`