Algebra - Factoring Trinomials

FACTORING IS THE REVERSE of multiplying. Skill in factoring, then, depends upon skill in multiplying: As for a quadratic trinomial --

2x² + 9x − 5

-- it will be factored as a product of binomials:

(? ?)(? ?)

Now, how will 2x² be produced? There is only one way: 2x• x :

(2x ?)(x ?)

And how will 5 be produced? Again, there is only one way: 1• 5. But does the 5 go with 2x or with x ?

(2x 5)(x 1) or (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• 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 x ?
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 = S:(2x 3)(x 5)

b) 2x² − 7x − 15 = S:(2x 3)(x 5)

c) 2x² − x − 15 = S:(2x 5)(x 3)

d) 2x² − 13x + 15 = S:(2x 3)(x 5)

Note: When the constant term is negative, as in parts a), b), c), then the signs in each factor will be different. But when that term is positive, as in part d), the signs will be the same. Usually, however, that happens by itself.

Nevertheless, can you correctly factor the following?

2x² − 5x + 3 = S:(2x − 3)(x − 1)

Problem 2 Factor these trinomials.

a) 3x² + 8x + 5 = S: ___________________

b) 3x² + 16x + 5 = S: _________________

c) 2x² + 9x + 7 = S:___________________

d) 2x² + 15x + 7 = S: __________________

e) 5x² + 8x + 3 = S:____________________

f) 5x² + 16x + 3 = S: ___________________

Problem 3. Factor these trinomials. The first one is done for you.

a) 2x² − 7x + 5 = S:(2x − 5)(x − 1)

b) 2x² − 11x + 5 = S: ______________________

c) 3x² + x − 10 = S:_____________________

d) 2x² − x − 3 = S:_______________________

e) 5x² − 13x + 6 = S:_____________________

f) 5x² − 17x + 6 = S:_______________________

g) 2x² + 5x − 3 = S:_______________________

h) 2x² − 5x − 3 = S:______________________

i) 2x² + x − 3 = S:________________________

j) 2x² − 13x + 21 = S:_________________________

k) 5x² − 7x − 6 = S:__________________________

i) 5x² − 22x + 21 = S:__________________________