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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.

The simplest algebraic identities are:

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An algebraic identity expresses the identity of two expressions that may look very different.

Unlike an equation, an identity is true for any values we may want to 'plug-in'.

For an identity, the left hand side of the equality and the right hand side of it are identical and may differ only in appearance.

Identities are used to shorten computations.

Many algebraic identities, such as the ones above, can be proven easily just by performing the operations involved and noticing how both sides are one and the same.

Example

Which number is larger?

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No solution is faster or simpler than by using identities.

By squaring both sides, we notice that 8 will occur on both sides whereas on the left hand side (LHS) there will be an additional positive term of the form 2ab. Therefore the left hand side is larger.

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Note that it is not necessary to compute the values in order to tell which one is larger.

Example

The sum and difference formula is a simple algebraic identity.

This is how we prove it:

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In solving problems, it is important to have a good ability to recognize patterns that belong to various identities. Perhaps, by replacing the pattern with the equivalent form of the expression, the problem becomes easier to solve.

Example

Compute without using a calculator the square of the number 999999.

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Application:

A very interesting and useful application of these identities is the identity:

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This identity is very useful in processing reciprocal polynomial equations, symmetric non-linear systems, as well as other algebraic expressions.

It is often used in the form:

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Using the difference, we can prove an interesting inequality:

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Since a square is always greater than or equal to zero:

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