Algebraic Identities: (x+y)³ & (x-y)³
The expressions shown above are called cubes of binomials.
Let's learn to expand these expressions:
(
x
+
y
)
3
(
x
−
y
)
3
The cube of binomial can be written as shown.
Since,
a
3
=
a
×
a
2
The square in RHS is expanded further by using the identity:
(
x
+
y
)
2
=
x
2
+
2
x
y
+
y
2
The product on RHS is evaluated using the distributive property.
Then the like terms in RHS can be added.
Thus,
(
x
+
y
)
3
=
x
3
+
3
x
2
y
+
3
x
y
2
+
y
3
Let’s verify this identity.
In the LHS of the identity,
we put
x
=
2
&
y
=
3
Similarly in the RHS of the identity,
we put
x
=
2
&
y
=
3
Thus, the identity is verified.
Now, let's expand the second Identity
If we replace
y
with
(
−
y
)
the expression changes to
(
x
−
y
)
3
So to find the expansion of
(
x
−
y
)
3
,
we can replace
y
with
(
−
y
)
in
(
x
+
y
)
3
=
x
2
+
3
x
2
y
+
3
x
y
2
+
y
3
This is the required expansion for
(
x
−
y
)
3
Let’s now use these identities to factorize polynomials.
To factorize this polynomial, it can be compared with the expansion of either
(
x
+
y
)
3
or
(
x
−
y
)
3
Since all the terms of the polynomial are positive, it is compared with the expansion of
(
x
+
y
)
3
Terms of polynomials are rearranged to compare with the terms in the identity.
Then terms that are perfect cubes are identified.
Comparing the polynomial with the identity we have,
x
=
3
a
&
y
=
b
Using the values of
x
&
y
, other terms of the polynomials are written as shown.
Since,
x
3
+
3
x
2
y
+
3
x
y
2
+
y
3
=
(
x
+
y
)
3
Let's factorize another polynomial.
This has both positive and negative terms, so it can be compared with the expansion of
(
x
−
y
)
3
The terms of polynomials are rearranged.
Then terms that are perfect cubes are identified.
Comparing the polynomial with the identity we have,
x
=
2
a
&
y
=
3
b
Using the values of
x
&
y
, other terms of the polynomials are written as shown.
Since
x
3
−
3
x
2
y
+
3
x
y
2
−
y
3
=
(
x
−
y
)
3
Revision
Expansion of cube of binomial
(
x
+
y
)
Replacing
y
with
(
−
y
)
in the identity,
(
x
+
y
)
3
=
x
3
+
3
x
2
y
+
3
x
y
2
+
y
3
The End