CHAPTER
8-6: FACTORING TRINOMIALS
We’ll factor them
and then put our FULLY FACTORED FORM on the same graph as the SIMPLIFIED FORM….WHAT SHOULD
HAPPEN IF WE FACTORED THE TRINOMIAL (quadratic, 2nd degree polynomial)
CORRECTLY?
FACTORING
TRINOMIALS WITH A POSITIVE SIGN AS THE SECOND SIGN:
You're trying to
turn a trinomial back to the two binomials that were multiplied together to get
it!
Always check your factoring by FOILing back!
FACTORING TRINOMIALS WITH A
PLUS SIGN AS
THE SECOND SIGN:
I have a simple method of foiling basic trinomials!
1. Set up your (
)( )
2. When the last sign is positive, then both signs in each of
the ( ) are the same!
3. How do you know what the 2 signs are? It's whatever
the sign is of the second (middle) term. Put
that sign in both parentheses.
4. To UNFOIL (factor), you will need to find
2 FACTORS that MULTIPLY to the LAST term, and ALSO ADD to the MIDDLE term.
To help you do this, I suggest you use and “X”
Put the product in the top of the X,
the sum in the
bottom of the X and
the correct
factors on the left and right of the X....
Understand that this is really just an educated guess and check!
EXAMPLE:
x2 + 8x +15
( )( )
THINK: Last sign is + so both signs are the same
THINK: Middle sign is + so both signs are +
( + )( + )
You know the the "F" in FOIL means that both first terms
must be x
( x + )( x + )
Now to get the "L" in FOIL, you need 2 factors whose product is 15
Like 1 and 15, or 3 and 5
But you also need to add to the I and O in FOIL which means that the two
factors must add to 8
____ x ____ = 15
____ + ____ = 8
Since 3 + 5 = 8, this must be the two factors that will work:
3 x 5 = 15
3 + 5 = 8
( x + 5 )( x + 3 )
(It doesn't matter
which factor you put in the first ( ) because they're the same
sign, but I always tend to put the larger number in the first parentheses for a
reason that you will see tomorrow)
Now FOIL to see if we're right!
NEXT EXAMPLE: LOOKS THE SAME WITH ONE DIFFERENCE...MIDDLE TERM IS NEGATIVE
x2 - 8x +15
(
)( )
THINK: Last sign is + so both signs are the same
THINK: Middle sign is - so both signs are -
( - )( - )
You know the the "F" in FOIL means that both first terms
must be x
( x - )( x - )
Now to get the "L" in FOIL, you need 2 factors whose product is 15
Like 1 and 15, or 3 and 5
But you also need to add to the I and O in FOIL which means that the two
factors must add to 8
-___ x -___ = 15
-___ + -___ = -8
Since -3 + -5 = -8, this must be the two factors that will work:
-3 x -5 = 15
-3 + -5 = -8
( x - 5 )( x - 3 )
(It doesn't matter
which is first because they're the same sign!)
Now FOIL to see if we're right!
I actually just
ignore the signs while making the educated guess because I've already put
negative signs in both parentheses so I've taken care of the negatives. It's up
to you which way you are most comfortable...we'll talk about that in class and
you'll try both ways...then YOU make up your mind which way you want to do it
and STAY with that method!
LAST EXAMPLE: SAME PROBLEM BUT NOW WITH A y ON THE MIDDLE AND LAST TERMS:
x2 - 8xy + 15y2
Simply use the same factorization as above and add the y:
( x - 5y)( x - 3y)
ALWAYS FOIL BACK
TO CHECK!!!
I'll also show you
"little smile" "big smile" to just check the middle
term....that's usually where you make a mistake if there is one....It
makes the check step shorter because you're just doing the "O" and
"I" in FOIL instead of the entire 4 multiplications.
FACTORING
TRINOMIALS WITH A NEGATIVE SIGN AS THE SECOND SIGN:
1. Set up your (
)( )
2. Look at the SECOND or last sign
If it's negative, then the signs in the ( ) must be
different
Why? Because when
you multiply integers and get a NEGATIVE product, the only way that will happen
is if they are DIFFERENT signs. Remember that the last term is the product of
the two LAST terms in FOILing.
3. Now look at the sign of the second term.
It tells you "Who wins," meaning which sign must have the bigger
number (absolute value)
Remember that the middle term is the SUM of the "O" and the
"I" terms when FOILing. Because these two terms have DIFFERENT
signs, when you add them, you actually "subtract" and take the bigger
number's sign.
Put that sign in
the first parentheses and always put the bigger number in the first
parentheses. (Although you can reverse the signs, but then just remember the
bigger factor goes with the sign of the 2nd term.
4. To UNFOIL
(factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but
SUBTRACT to the middle term. (yesterday the factors needed to ADD to the
middle)
Or you can still say you're adding, but since they are DIFFERENT signs, you
will end up subtracting!
Again, this is still an educated guess and check!
EXAMPLE:
x2 + 2x -15
( )( )
THINK: Last sign is - so signs are DIFFERENT!
THINK: Middle sign is + so POSITIVE MUST WIN
( + )( - )
You know the the "F" in FOIL means that both first terms
must be x
( x + )( x - )
Now to get the "L" in FOIL, you need 2 factors whose product
is NEGATIVE 15
Like -1 and 15, or -3 and 5, or 1 and -15, or 3 and -5
But since the POSITIVE must win according to the middle term of POSITIVE 2x,
you know that the bigger factor must be POSITIVE (so it can win!)
Therefore, your choices are POSITIVE 15 and -1 or POSITIVE 5 and -3
But you also need to add to the I and O in FOIL so pick the two factors
that also add to
POSITIVE 2
Actually, I think to myself the two factors that SUBTRACT to 2, since when you
add opposite signs, you don't add, you take the difference and take the higher
number's sign.
Since -3 + 5 = +2, this must be the two factors that will work.
So adding it would
look like this:
+____ x -____ = -15
+____ + -____ = + 2
so
5 x (-3) = -15
5 +( -3) = 2
( x + 5 )( x - 3 )
NEXT EXAMPLE: Same as before except the middle sign is now negative
x2 - 2x -15
( )( )
THINK: Last sign is - so signs are DIFFERENT!
THINK: Middle sign is - so NEGATIVE MUST WIN
that also add to POSITIVE 2
Actually, I think to myself the two factors that SUBTRACT to 2, since when you
add opposite signs, you don't add, you take the difference and take the higher
number's sign.
Since -3 + 5 = +2, this must be the two factors that will work.
I set it up like this:
____ x ____ = 15
____ - ____ = 2
so
5 x 3 = 15
5 - 3 = 2
USING ADDING IT WOULD LOOK LIKE THIS:
+____ x -____ = -15
+____ + -____ = + 2
so
5 x (-3) = -15
5 +( -3) = 2
( x + 5 )( x - 3 )
LAST EXAMPLE: Same as the last example but with a “y”:
x2 - 2xy -15y2
Same problem as the one before, except now there are 2 variables!
Simply use the same factorization and add the y
( x - 5y )( x + 3y)
