Note: NYS Integrated
Algebra requires the factoring of only trinomials whose leading
coefficients are 1.
If you need to deal with leading coefficients other than one, go to
Algebra 2.
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for this lesson
a will
always be 1.
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Whether
we use the distributive process, use FOIL, or line up the factors vertically to multiply,
we all know that:
The
expression
is called a
quadratic trinomial.
To factor a trinomial of this form, we need to reverse the multiplication
process we used above.
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ATTENTION
Super Sleuths:
We are on the hunt for factors!
There are many different ways to think about this process of "reversing"
multiplication. Let's look first at what is happening
and then at a shortcut process for finding the factors.
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Let's see what
is involved with factoring
 .
| 1. |
To get the
leading term of x², each first term will be x. So we start
with:
(x
) (x )
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| 2. |
The product of the
last terms must be -6. Unfortunately, we are now faced
with options, as there are several ways to arrive at a product
of -6:
+6 and -1
-6 and +1
+3 and -2
-3 and +2 |
All of these options will give us a product of -6. |
These different options make it "appear" that
we have several possible answers:
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(x + 6)(x - 1)
(x - 6)(x + 1) (x +
3)(x - 2)
(x - 3)(x + 2) |
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| 3.
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The possible answers created from our options
above do not ALL give us the correct result. We need to
find the combination that will yield
the correct "middle term" of +x (for this
problem).
(x + 6)(x - 1) gives
middle term 5x.
(x - 6)(x + 1) gives
middle term -5x.
(x + 3)(x - 2) gives
middle term +x. YEA!!!!!
(x - 3)(x + 2) gives
middle term -x.
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| 4. |
Answer:
Notice: While we initially had
several options for answers, we really had only one true answer.
The more options that a problem creates, the more detective work needed to find
the true answer. |
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SHORT CUT
PROCESS:
If the
coefficient of x2 is 1,
then x2 + bx + c = (x + m)(x
+ n)
where m and n multiply to give c
and m and n add to give b. |
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When the leading
coefficient is 1,
ask yourself
"what numbers multiply to the last term
and add to the middle term?"
In the example above,
,
you need numbers that multiply to -6 and add to +1.
These numbers will be +3 and -2 and create and answer of: (x
+ 3) (x - 2)
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