Which of the following is not an identity?
To determine which of the given expressions is not an identity, we’ll need to verify if the expansion of each algebraic expression on the left-hand side equals the expression on the right-hand side.
A) ((x + 1) \cdot (x + 1) = x^2 + 2x + 1)
Let’s expand:
[
(x + 1) \cdot (x + 1) = x^2 + x + x + 1 = x^2 + 2x + 1
]
This is an identity.
B) (a \cdot (a + 3) = a^2 + 3a)
Expand the left-hand side:
[
a \cdot (a + 3) = a^2 + 3a
]
This is also an identity.
C) ((m - 5) \cdot (m + 5) = m^2 - 5m + 25)
Perform the expansion on the left side:
[
(m - 5) \cdot (m + 5) = m^2 + 5m - 5m - 25 = m^2 - 25
]
Here, it’s clear that the expression simplifies to (m^2 - 25), which is different from (m^2 - 5m + 25). So, this is not an identity.
D) ((n - 2) \cdot (n - 2) = n^2 - 4n + 4)
Expand the left side:
[
(n - 2) \cdot (n - 2) = n^2 - 2n - 2n + 4 = n^2 - 4n + 4
]
This is an identity.
Hence, the expression which is not an identity is option C: ((m - 5) \cdot (m + 5) = m^2 - 5m + 25). The correct expanded form should be (m^2 - 25).