#
Section 6-6

#
Multiplying Polynomials: Special Cases (Perfect Squares, Product of a Sum and
a Difference)

There are two special cases when multiplying polynomials, which have easy
patterns you should memorize. These are the
square of a binomial
pattern
and the
**
product of a sum and a difference
**
pattern.

SQUARE OF A BINOMIAL

This is the special case in which you are asked to multiply

**
(
***
x
*
+
*
a
*
)(
*
x
*
+
*
a
*
).

Using
FOIL
, we get:

**
=
***
x
*
^{
2
}
+
*
ax + ax
*
*
+ a
*
^{
2
}

**
=
***
x
*
^{
2
}
+ 2
*
ax + a
*
^{
2
}

Example:

Multiply:

**
(
***
p
*
– 6)(
*
p
*
– 6)

Using the binomial square pattern with
*
a
*
=
**
–
**
6, we get:

**
(
***
p
*
– 6)(
*
p
*
– 6) =
*
p
*
^{
2
}
+ 2(–6
*
p
*
)
*
+
*
(–6)
^{
2
}

**
=
***
p
*
^{
2
}
– 12
*
p
*
*
+
*
36

PRODUCT OF A SUM AND A DIFFERENCE

The product of a sum and a difference of the same two expressions can be written
as a
difference of squares
.

**
(
***
x
*
+
*
a
*
)(
*
x
*
–
*
a
*
)

Using
FOIL
, we get

*
***
x
**
**
**^{
2
}
*
***
– ax
**
**
+
***
ax
*
–
*
a
*
^{
2
}

**
=
***
x
*
^{
2
}
–
*
a
*
^{
2
}

Example:

Multiply:

**
(
***
r
*
^{
3
}
+
*
s
*
)(
*
r
*
^{
3
}
–
*
s
*
)

Using the product of a sum and a difference pattern, we get:

**
(
***
r
*
^{
3
}
+
*
s
*
)(
*
r
*
^{
3
}
–
*
s
*
)

**
=
***
r
*
^{
6
}
–
*
s
*
^{
2
}