- The
general rule is that each
term in the first factor
has to multiply each term
in the other factor
- The
number of products you get
has to be the number of
terms in the first factor
times the number of terms
in the second factor. For
example, a binomial times
a binomial gives four products,
while a binomial times a
trinomial gives six products.
- Be
very careful and methodical
to avoid missing any terms
- After
the multiplication is complete
you can try to collect like
terms to simplify the result
(x + 2)(x2- 2x + 3)
There are six possible products.
We can start with the x
and multiply it by all three
terms in the other factor,
and then do the same with
the 2. It would look like
this:
(x + 2)(x2-
2x + 3)
= (x)x2-(x)2x + (x)3 + (2)x2-(2)2x + (2)3
= x3-
2x2
+ 3x + 2x2- 4x + 6
= x3-
x + 6
This method can get hard to keep
track of when there are many
terms. There is, however,
a more systematic method based
on the stacked method of multiplying
numbers:
|
Stack
the factors, keeping
like degree terms lined
up vertically: |
|
|
Multiply
the 2 and the 3: |
|
|
Multiply
the 2 and the –2x: |
|
|
Multiply
the 2 and the x2: |
|
|
Now
multiply the x
by each term above it,
and write the results
down underneath, keeping
like degree terms lined
up vertically: |
|
|
|
|
|
|
|
|
Then
you just add up the
like terms that are
conveniently stacked
above one another: |
|
This stacked method is much safer, because you are far less
likely to accidentally overlook
one of the products, but it
does take up more space on
the paper.
Example:
ab(2a + 1) = ab(2a) + ab(1) =
2a2b + ab
Because
the situation of a binomial
times a binomial is so common,
it helps to use a quick mnemonic
device to help remember all
the products. This is called
the FOIL method.
Example:
1.
The F stands for first,
which means the x in
the first factor times the
x in the second factor
2.
The O stands for outer,
which means the x in
the first factor times the
3 in the second factor
3.
The I stands for inner,
which means the 2 in the first
factor times the x
in the second factor
4.
The L stands for last,
which means the 2 in the first
factor times the 3 in the
second factor
·
Of course you
would then combine the 3x
+ 2x into a 5x,
because they are like terms,
so the final result is
(x
+ 2)(x + 3) = x2
+ 5x + 6 |
