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

1 Answer
Mar 21, 2018

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:
enter image source here

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)