In this lesson we will learn how to factor a trinomial
of the form: ax² + bx + c,
 where "a" is the coefficient of the "x²" term,
"b" is the coefficient of the "x" term,
and "c" will always be a constant.
In this lesson "a" will always be 1.
  Only trinomials written in this form ax² + bx + c
can be factored using the method we will learn in this lesson.
First, let's look at some examples of trinomials
which have been written in the correct form:
x² + 5x +6
(a = 1, b = 5, c = 6)
x² + 10x + 25
(a = 1, b = 10, c = 25)
x² - 3x - 28
(a = 1, b = -3, c = -28)

The factors will always be 2 binomials which have these characteristics:
1.  The product of the first terms in each binomial
must be equal to the first term in the trinomial (x²).
2.  The product of the last terms in each binomial
must be equal to the last term in the trinomial (c).
3.  The sum of the last terms in the binomials must
ADD up to the coefficient of the middle term in the trinomial (b).
 

Factor x² + 3x + 2

1.  The product of the first term in both binomials must be x².
That means that each first term must be "x".
x² + 3x + 2 = (x     )(x    )
2.  The product of the last terms in each binomial must be +2.
However, they must also add up to the
coefficient of the middle term (+3). 
Since the only factors of +2 are (2)(1), and since (2) + (1) = +3
The last terms in the binomials must be (2) and (1).
So the solution is:
x² + 3x + 2 = (x + 2)(x + 1)

Factor a² - 9a + 14
The first term in each binomial of the answer is easy.....
only (a)(a) = a²....so the solution begins as:
(a    )(a    )
The last terms are a little tricky....
Remember, they must multiply together to give you
the constant term (+14), AND add up to the coefficient
of the middle term (-9).
If the factors which meet both of those conditions don't come popping
into your head...a very good strategy is to list all the possibilities.
Let's look at all the possible factors of +14
(+14)(+1)
(-14)(-1)
(+7)(+2)
(-7)(-2)
Remember....not only must they multiply together
to give you +14, they must ADD up to -9!
Can you find the correct pair of factors from the list above?
That's correct!   Only (-7) and (-2) will multiply to give you +14,
but they also add up to -9!!
So the final answer to our factoring problem is:
a² - 9a + 14 = (a - 7)(a - 2)

Factor c² + 5c - 24
Again...the first term in each binomial is easy
(c      )(c     )
List the factors of -24:
(-24)(1)
(24)(-1)
(-12)(2)
(12)(-2)
(-8)(3)
(8)(-3)
product is -24, sum is +5
(-6)(4)
(6)(-4)
So the solution is:
c² + 5c -24 = (c+8)(c-3)
 

Factor z² + 16z -17
(z    )(z    )...easy!!!
Factors of -17 which add up to +16 are (+17) and (-1)
So the solution is:
z² + 16z - 17 = (z + 17)(z - 1)

To practice factoring trinomials,
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