Factoring Trinomials
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: 5x^{2}  22x + 8
Solution
Since the first term of the trinomial is 5x^{2}, the product of the first terms of
the binomials is 5x^{2}. Therefore, it is reasonable to guess that the first terms
are 5x and 1x.
Thus, 5x^{2}  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 5x^{2}  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.
Pissibilities 
F O I L 
Resulting trinomial 
(5x  8)(1x  1)
(5x  4)(1x  2)
(5x  1)(1x  8)
(5x  2)(1x  4) 
= 5x^{2}  5x  8x + 8
= 5x^{2}  10x  4x + 8
= 5x^{2}  40x  1x + 8
= 5x^{2}  20x  2x + 8 
= 5x^{2}  13x + 8
= 5x^{2}  14x + 8
= 5x^{2}  41x + 8
= 5x^{2}  22x + 8 
The last guess yields the correct middle term, 22x.
So, the factorization is (5x  2)(x  4).
