Factoring: Trial and Error
After we have gained some experience at factoring by grouping, we can often find
the factors without going through the steps of grouping. Consider the polynomial
2x2 - 7x + 6.
The factors of 2x2 can only be 2x and x. The factors of 6 could be 2 and 3 or 1
and 6. We can list all of the possibilities that give the correct first and last terms
without putting in the signs:
| (2x 2)(x 3) |
(2x 6)(x 1) |
| (2x 3)(x 2) |
(2x 1)(x 6) |
Before actually trying these out, we make an important observation. If (2x 2) or
(2x 6) were one of the factors, then there would be a common factor 2 in the original
trinomial, but there is not. If the original trinomial has no common factor, there
can be no common factor in either of its linear factors. Since 6 is positive and the
middle term is -7x, both of the missing signs must be negative. So the only possibilities
are (2x - 1)(x - 6) and (2x - 3)(x - 2). The middle term of the first product
is -13x, and the middle term of the second product is -7x. So we have found
the factors:
2x2 - 7x + 6 = (2x - 3)(x - 2)
Even though there may be many possibilities in some factoring problems, often
we find the correct factors without writing down every possibility. We can use a bit
of guesswork in factoring trinomials. Try whichever possibility you think might
work. Check it by multiplying. If it is not right, then try again. That is why this
method is called trial and error.
Example 1
Trial and error
Factor each quadratic trinomial using trial and error.
a) 2x2 + 5x - 3
b) 3x2 - 11x + 6
Solution
a) Because 2x2 factors only as 2x · x and 3 factors only as 1 · 3, there are only two
possible ways to factor this trinomial to get the correct first and last terms:
(2x 1)(x 3) and (2x 3)(x 1)
Because the last term of the trinomial is negative, one of the missing signs must
be +, and the other must be -. Now we try the various possibilities until we get
the correct middle term:
(2x + 1)(x - 3) = 2x2 - 5x - 3
(2x + 3)(x - 1) = 2x2 + x - 3
(2x - 1)(x + 3) = 2x2 + 5x - 3
Since the last product has the correct middle term, the trinomial is factored as
2x2 + 5x - 3 = (2x - 1)(x + 3).
b) There are four possible ways to factor 3x2 - 11x + 6:
| (3x 1)(x 6) |
(3x 2)(x 3) |
| (3x 6)(x 1) |
(3x 3)(x 2) |
Because the last term is positive and the middle term is negative, both signs must
be negative. Now try possible factors until we get the correct middle term:
(3x - 1)(x - 6) = 3x2 - 19x + 6
(3x - 2)(x - 3) = 3x2 - 11x + 6
The trinomial is factored correctly as
3x2 - 11x + 6 = (3x - 2)(x - 3).