Algebraic Identities

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Introduction to Algebraic Identities

Let us say that we are dealing with two equations. The two equations are:

\({\,\,5x \,– 3 = 12}\)

\({10x \,– 6 = 24}\)

If you solve both equations separately, you will see that the value of \({x}\) is 3 in both cases. If you write the equations in the form \({ax \,– b = c,}\) then you will see that the two equations are:

\({ax\, – b = c}\)

\({2(ax \,– b) = 2c}\)

What you did there was notice a pattern. Allowing us to observe patterns is how algebra appears in our daily lives.

The Big Idea: Algebraic Identities

A simple idea: What are identities?

Simple patterns that allow us to simplify complicated algebraic equations into manageable bits are called identities.

As discussed earlier, algebraic expressions and equations are a means to express language mathematically. Just how language has special terms that appear very frequently in basic sentence construction, like nouns, prepositions, articles, etc., algebra has identities.

These identities are commonly occurring patterns that help in simplifying a convoluted mess of \({x}\)’s and \({y}\)’s to something that looks presentable and can be understood by any reader.

Here’s a nice way to visualize a common algebraic identity:

List of algebraic identities

This is the list of the most useful and commonly used algebraic identities:

\(\begin{align}   & 1.\quad{\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}\\   & 2.\quad{\left( {x - y} \right)^2} = {x^2} - 2xy + {y^2}\\    & 3.\quad\left( {x + y} \right).\left( {x - y} \right) = {x^2} - {y^2}\\    & 4.\quad\left( {x + a} \right).\left( {x + b} \right) = {x^2} + x\left( {a + b} \right) + ab\\    & 5.\quad\left( {x + a} \right).\left( {x - b} \right) = {x^2} + x\left( {a - b} \right)-ab\\    & 6.\quad\left( {x - a} \right).\left( {x + b} \right) = {x^2} + x\left( {b-a} \right)-ab\\    & 7.\quad\left( {x - a} \right).\left( {x - b} \right) = {x^2}-x\left( {a + b} \right) + ab   \end{align}\)
\(\begin{align}   & 8.\quad{\left( {x + y} \right)^3} = {a^3} + {b^3} + 3xy\left( {x + y} \right)\\    & 9.\quad{\left( {x-y} \right)^3} = {a^3}-{b^3} - 3xy\left( {x-y} \right)\\    & 10.\;\;\,{\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2xz\\    & 11.\;\;\,{\left( {x + y-z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy-2yz-2xz\\    & 12.\;\;\,{\left( {x-y + z} \right)^2} = {x^2} + {y^2} + {z^2}-2xy + 2yz-2xz\\    & 13.\;\;\,{\left( {x-y-z} \right)^2} = {x^2} + {y^2} + {z^2}-2xy + 2yz-2xz    \end{align}\)

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