In this article, we will cover 3 important algebraic identities with examples. Algebraic identities are following:
1.
2.
3.
Let’s go through first algebraic identity.
.
It means that expression of the form
is equal to
.
Let’s take an example, we have
. Comparing
with
we get,
and
.
By algebraic identity 1, we know that
.
Therefore,
.
Similarly,
.
This is not just about expanding the expressions which are of the form
but it is also about factorizing the polynomials which are of the form
.
GMAT Sample Problem 1:
is equal to:
(A)
(B)
(C)
(D) None of the above
which is of the form
.
By algebraic identity 1, we know that
.
Therefore,
.
Hence, the answer is (B).
Now, we will cover sample problem regarding algebraic identity 2. According to algebraic identity 2, we have:
GMAT Sample Problem 2:
is equal to:
(A)
(B)
(C)
(D) None of the above
which is of the form
.
By algebraic identity 2, we know that
.
Therefore,
Hence, the answer is (A).
Let’s learn about 3rd algebraic identity now. According to 3rd algebraic identity we have:
.
Example:
GMAT Sample Problem 3:
is equal to:
(A)
(B)
(C)
(D) None of the above
which is of the form
.
By algebraic identity 3, we know that
.
Therefore,
.
Hence, the answer is (B).
Now we can try some complicated problems regarding these algebraic identities because they are important considering GMAT exam.
GMAT Sample Problem 4:
is equal to:
(A)
(B)
(C)
(D)
(E) None of the above
We can write expression
as
which is of the form
.
By algebraic identity 3, we know that
.
Therefore,
(1)
Similarly, we have
(2)
Putting equation (2) in equation (1) we get,
can be further expanded to
.
Therefore,
Hence, the answer is (C).
GMAT Sample Problem 5:
is equal to:
(A)
(B)
(C)
(D)
(E) None of the above
can be written as
which is of the form
.
By algebraic identity 1, we know that
.
Therefore,
Hence, the answer is (A).
GMAT Sample Problem 6:
is equal to:
(A)
(B)
(C)
(D) None of the above
can be written as
which is of the form
.
By algebraic identity 2, we know that
.
Therefore,
Hence, the answer is (B).
After good amount of practice, you would be able to answer similar problems by just looking at the choices. Let’s try one more problem.
GMAT Sample Problem 7:
is equal to:
(A)
(B)
(C)
(D)
can be written as
which is of the form
By algebraic identity 3, we know that
Therefore,
(1)
Similarly, we have
(2)
Putting equation (2) in equation (1), we get
Hence, the answer is (C).
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