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Multiplication of polynomials
Some terminology
.
Consider the
expression
+

+
+
a
2
b
2
a c
5
a
3
d
a c d
3
3
7 ,
where
a
,
b
,
c
and
d
are (unknown) real numbers.
a
,
b
,
c
and
d
are the
variables
in the expression.
This expression is the sum of the three
terms
:
a
2
b
,
2
a c
,

5
a
3
d
,
a c d
3
,
3
7.
Each of these terms has a numerical factor called the
coefficient
of the term
The coefficient of

5
a
3
d
is

5 while the coefficient of
the term
a c d
3
is
1
3
.
3
7 is a
constant term
as it does not contain a variable.
Since each term in the expression is a constant coefficient times a product of variables raised to a positive integer power, it is a
polynomial
in the variables that occur.
A polynomial with exactly two terms is called a
binomial
, while a polynomial with three terms is called a
trinomial
.
Multiplication of polynomials
.
Multiplication of polynomials can be achieved by using the distributive property of multiplication over addition.
For example, two binomials
+
a
b
and
+
c
d
can be multiplied as follows.
=
(
)
+
a
b
(
)
+
c
d
+
(
)
+
a
b c
(
)
+
a
b d
=
+
+
+
a c
b c
a d
b d
.
The first step uses distributivity of multiplication over addition on the left. In more detail, if
=
S
+
a
b
.
=
(
)
+
a
b
(
)
+
c
d
S .
(
)
+
c
d
=
+
S . c
S . d
=
+
(
)
+
a
b c
(
)
+
a
b d
.
Alternatively, we can start by using distributivity of multiplication over addition on the right.
=
(
)
+
a
b
(
)
+
c
d
+
a
(
)
+
c
d
b
(
)
+
c
d
=
+
+
+
a c
a d
b c
b d
.
In more detail, if
=
T
+
c
d
, the first step is:
=
(
)
+
a
b
(
)
+
c
d
(
)
+
a
b . T
=
+
a . T
b . T
=
+
a
(
)
+
c
d
b
(
)
+
c
d
.
The first term term
a c
in the product is obtained by multiplying the
first
(F) terms
a
and
c
of the two binomials and the last term
b d
in
the product is obtained by multiplying the
last
(L) terms
b
and
d
of the two binomials.
The term
a d
is obtained by multiplying the
"
outer
" (O) terms
a
and
d
together while the term
b c
is obtained by multiplying the "
inner
" (I) terms
b
and
c
together.
The memnonic
FOIL
, with each letter standing for one of the four terms in the product, can be used to help remember how to multiply
binomials.
If the variables
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