What is a Polynomial Identity?

What is a Polynomial Identity?

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Consider the following relation:

\({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\)

This is true for every value of a and b, and hence this is an example of an identity. In simple words, identities are equalities which are always true. Here is another example of an identity (which is related to the one above):

\[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]

Example 1: Evaluate \({1001^2}\) using the first identity above.

Solution: We have

\[\begin{align}&{1001^2} = {\left( {1000 + 1} \right)^2}\\&\;\;\;\;\;\;\;\;\; = {1000^2} + 2\left( {1000} \right)\left( 1 \right) + {1^2}\\&\;\;\;\;\;\;\;\;\;= 1000000 + 2000 + 1\\&\;\;\;\;\;\;\;\;\;= 1002001\end{align}\]

Example 2: Evaluate \({998^2}\)using the second identity above.

Solution: We have

\[\begin{array}{l}{998^2} = {\left( {1000 - 2} \right)^2}\\\;\;\;\;\;\;\;\; = {1000^2} - 2\left( {1000} \right)\left( 2 \right) + {2^2}\\\;\;\;\;\;\;\;\; = 1000000 - 4000 + 4\\\;\;\;\;\;\;\;\; = 996004\end{array}\]

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