Question

give me examples and answers to factoring expressions such as 12x^2-9x+15

Answer 1

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)
Note: When the constant term is negative, as in parts a), b), c), then the signs in each factor must be different. But when the constant term is positive, as in part d), the signs must be the same. Usually, however, that happens by itself.
Nevertheless, can you correctly factor the following?
2x² − 5x + 3 = (2x − 3)(x − 1)
Problem 2. Factor these trinomials.
a) 3x² + 8x + 5 = (3x + 5)(x + 1)
b) 3x² + 16x + 5 = (3x + 1)(x + 5)
c) 2x² + 9x + 7 = (2x + 7)(x + 1)
d) 2x² + 15x + 7 = (2x + 1)(x + 7)
e) 5x² + 8x + 3 = (5x + 3)(x + 1)
f) 5x² + 16x + 3 = (5x + 1)(x + 3)
Problem 3. Factor these trinomials.
a) 2x² − 7x + 5 = (2x − 5)(x − 1)
b) 2x² − 11x + 5 = (2x − 1)(x − 5)
c) 3x² + x − 10 = (3x − 5)(x + 2 )
d) 2x² − x − 3 = (2x − 3)(x + 1)
e) 5x² − 13x + 6 = (5x − 3)(x − 2)
f) 5x² − 17x + 6 = (5x − 2)(x − 3)
g) 2x² + 5x − 3 = (2x − 1)(x + 3)
h) 2x² − 5x − 3 = (2x + 1)(x − 3)
i) 2x² + x − 3 = (2x + 3)(x − 1)
j) 2x² − 13x + 21 = (2x − 7 )(x −3)
k) 5x² − 7x − 6 = (5x + 3)(x − 2)
i) 5x² − 22x + 21 = (5x − 7)(x − 3)