- 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:
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Stack the factors, keeping like degree terms lined up vertically:
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Multiply the 2 and the 3:
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Multiply the 2 and the –2x:
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Multiply the 2 and the x2:
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Now multiply the x by each term above it, and
write the results down underneath, keeping like degree terms lined up
vertically:
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Then you just add up the like terms that are
conveniently stacked above one another:
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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
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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
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