Question

How do you factor the expression `35x^3 - 19x^2 + 2x`?

Answer 1

x(7x - 1)(5x - 2)

Explanation:

`f(x) = xy = x(35x^2 - 19x + 2)`
Factor y, the trinomial in parentheses, by the new AC Method (Socratic Search).
`y = 35x^2 - 19x + 2 =` 35(x + p)(x + q)
Converted trinomial: `y' = x^2 - 19x + 70 =` (x + p')(x + q').
p' and q' have same sign because ac > 0.
Factor pairs of (ac = 70) --> (-2, -35)(-5, -14). This sum is (-19 = b). Then, p' = -5 and q' = -14.
Back to trinomial y --> `p = (p')/a = -5/35 = -1/7`, and
`q = (q')/a = -14/35 = -2/5`
Factored form:
`y = 35(x - 1/7)(x - 2/5) = (7x - 1)(5x - 2)`.
Finally:
`f(x) = x(7x - 1)(5x - 2)`