FACTORING IS THE REVERSE of multiplying. Skill in factoring, then, depends upon skill in multiplying: Lesson 16. As for a quadratic trinomial --
2x² + 9x − 5
-- it will be factored as a product of binomials:
(? ?)(? ?)
The first term of each binomial will be the factors of 2x², and the second term will be the factors of 5.
Now, how can we produce 2x²? There is only one way: 2x· x :
(2x ?)(x ?)
And how can we produce 5? Again, there is only one way: 1· 5. But does the 5 go with 2x --
(2x 5)(x 1)
or with x --
(2x 1)(x 5) ?
Notice: We have not yet placed any signs
How shall we decide between these two possibilities? It is the combination that will correctly give the middle term, 9x :
2x² + 9x − 5.
Consider the first possibility:
(2x 5)(x 1)
Is it possible to produce 9x by combining the outers and the inners:
2x (that is, 2x· 1) with 5x ?
No, it is not. Therefore, we must eliminate that possibility and consider the other:
(2x 1)(x 5)
Can we produce 9x by combining 10x with 1x ?
Yes -- if we choose +5 and −1:
(2x − 1)(x + 5)
(2x − 1)(x + 5) = 2x² + 9x − 5.
Skill in factoring depends on skill in multiplying -- particularly in picking out the middle term
Problem 1. Place the correct signs to give the middle term.
a) 2x² + 7x − 15 = (2x − 3)(x + 5)
b) 2x² − 7x − 15 = (2x + 3)(x − 5)
c) 2x² − x − 15 = (2x + 5)(x − 3)
d) 2x² − 13x + 15 = (2x − 3)(x − 5)