We can also use trial and error to try to factor a trinomial. In this method,
we make educated guesses about the factors. Then we multiply to see if
one of the guesses is correct.
Factor by trial and error: 5x2 - 22x + 8
Since the first term of the trinomial is 5x2, the product of the first terms of
the binomials is 5x2. Therefore, it is reasonable to guess that the first terms
are 5x and 1x.
Thus, 5x2 - 22x + 8 = (5x + ?)(1x + ?)
Because the last term of the trinomial is 8, the product of the last terms of
each binomial must also be 8. The possible pairs of integers whose product
is 8 are 8 Â· 1, 4
Â· 2, -8 Â· (-1), and
-4 Â· (-2).
Since the last term of 5x2 - 22x + 8 is positive and the middle term is
negative, we need only check the negative integer factors of 8.
We can use the FOIL method to check each possibility.
||F O I L
|(5x - 8)(1x - 1)
(5x - 4)(1x - 2)
(5x - 1)(1x - 8)
(5x - 2)(1x - 4)
| = 5x2 - 5x - 8x + 8
= 5x2 - 10x - 4x + 8
= 5x2 - 40x - 1x + 8
= 5x2 - 20x - 2x + 8
= 5x2 - 13x + 8
= 5x2 - 14x + 8
= 5x2 - 41x + 8
= 5x2 - 22x + 8
The last guess yields the correct middle term, -22x.
So, the factorization is (5x - 2)(x - 4).