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Byju's Answer
Standard IX
Mathematics
Algebraic Identities
Verify x3 -...
Question
Verify
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
using some non-zero positive integers and check by actual multiplication. Can you call theses as identities?
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Solution
To prove:
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
Consider the right hand side (
R
H
S
) and expand it as follows:
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
=
x
3
+
x
2
y
+
x
y
2
−
y
x
2
−
x
y
2
−
y
3
=
(
x
3
−
y
3
)
+
(
x
2
y
+
x
y
2
+
x
2
y
−
x
y
2
)
=
x
3
−
y
3
=
L
H
S
Hence proved.
Yes, we can call it as an identity: For example:
Let us take
x
=
2
and
y
=
1
in
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
then the
L
H
S
and
R
H
S
will be equal as shown below:
2
3
−
1
3
=
7
and
(
2
−
1
)
(
2
2
+
(
2
×
1
)
+
1
2
)
=
1
(
5
+
2
)
=
1
×
7
=
7
Therefore,
L
H
S
=
R
H
S
Hence,
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
can be used as an identity.
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Similar questions
Q.
Verify the relation,
x
3
+
y
3
=
(
x
+
y
)
(
x
2
−
x
y
+
y
2
)
using some non-zero positive integers and check by actual multiplication. Can you call this equation an identity?
Q.
Verify:
(i)
x
3
+
y
3
=
(
x
+
y
)
(
x
2
−
x
y
+
y
2
)
(ii)
x
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−
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3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
Q.
Verify (i)
x
3
+
y
3
=
(
x
+
y
)
(
x
2
+
x
y
+
y
2
)
Q.
Check:
x
3
−
y
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=
(
x
−
y
)
(
x
2
+
x
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2
)
Q.
x
3
+
y
3
=
(
x
+
y
)
(
x
2
−
x
y
+
y
2
)
prove this identity.
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