Algebraic identities are equalities which remain true regardless of the values of any variables which appear within it.

In our website, we have provided two calculators for algebraic identities.

One is to find the expansion for (a + b)^{n} and other one is to find the expansion for (a - b)^{n}.

Please click the below links to get the linear regression needed.

**Expansion Calculator for (a + b) ^{n}**

**Expansion Calculator for (a - b) ^{n}**

If you would like to have problems on algebraic identities, please click the link given below.

**Worksheet on Algebraic Identities**

In this section, we are going to see, how to prove the expansions of algebraic identities geometrically.

Let us consider algebraic identity and its expansion given below.

(a + b)^{2} = a^{2} + 2ab + b^{2}

We can prove the the expansion of (a + b)^{2} using the area of a square as shown below.

In this section, we are going to see the list of identities which are being used to solve all kind of problems in the Algebra.

(a + b) (a + b) | |

(a - b) (a - b) | |

a | |

(x + a)(x + b) = x |
Examples |

(a + b) (a + b) | |

(a - b) (a - b) | |

a a | |

a a |

(a + b + c)^{2 }= a^{2 }+ b^{2 }+ c^{2} + 2ab + 2bc + 2ac

(a + b - c)^{2 }= a^{2 }+ b^{2 }+ c^{2} + 2ab - 2bc - 2ac

(a - b + c)^{2 }= a^{2 }+ b^{2 }+ c^{2} - 2ab - 2bc + 2ac

(a - b - c)^{2 }= a^{2 }+ b^{2 }+ c^{2} - 2ab + 2bc - 2ac

a^{2} + b^{2} = (a + b)^{2} - 2ab

a^{2} + b^{2} = (a - b)^{2} + 2ab

a^{2} + b^{2} = 1/2 ⋅ [(a + b)^{2} + (a - b)^{2}]

ab = 1/4 ⋅ [(a + b)^{2} - (a - b)^{2}]

(a + b + c)^{3} = a^{3} + b^{3} + c^{3} + 3a^{2}b + 3a^{2}c + 3ab^{2} + 3b^{2}c + 3ac^{32} + 3bc^{2} + 6abc

(a + b - c)^{3} = a^{3} + b^{3} - c^{3} + 3a^{2}b - 3a^{2}c + 3ab^{2} - 3b^{2}c + 3ac^{2} + 3bc^{2} - 6abc

(a - b + c)^{3} = a^{3} - b^{3} + c^{3} - 3a^{2}b + 3a^{2}c + 3ab^{2} + 3b^{2}c + 3ac^{2} - 3bc^{2} - 6abc

(a - b - c)^{3} = a^{3} - b^{3} - c^{3} - 3a^{2}b - 3a^{2}c + 3ab^{2} - 3b^{2}c + 3ac^{2} - 3bc^{2} + 6abc

We can remember algebraic identities expansions like

(a + b)^{2}, (a + b + c)^{2}, (a + b + c)^{3}

In the above identities, if one or more terms is negative, how can we remember the expansion ?

This question has been answered in the following three cases.

**Case 1 :**

For example, let us consider the identity of (a + b + c)^{2}

We can easily remember the expansion of (a + b + c)^{2}.

If c is negative, then we will have

(a + b - c)^{2}

How can we remember the expansion of (a + b - c)^{2} ?

It is very simple.

Let us consider the expansion of (a + b + c)^{2}_{.}

**(a + b + c) ^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca**

In the terms of the expansion above, consider the terms in which we find "c".

They are c^{2}, bc, ca.

Even if we take negative sign for "c" in c^{2}, the sign of c^{2} will be positive. Because it has even power 2.

The terms bc, ca will be negative. Because both "b" and "a" are multiplied by "c" that is negative.

Finally, we have

**(a + b - c) ^{2} = a^{2} + b^{2} + c^{2} + 2ab - 2bc - 2ca**

**Case 2 :**

In (a + b + c)^{2}, if "b" is negative, then we will have

(a - b + c)^{2}

How can we remember the expansion of (a - b + c)^{2} ?

It is very simple.

Let us consider the expansion of (a + b + c)^{2}_{.}

**(a + b + c) ^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca**

In the terms of the expansion above, consider the terms in which we find "b".

They are b^{2}, ab, bc.

Even if we take negative sign for "b" in b^{2}, the sign of b^{2} will be positive. Because it has even power 2.

The terms ab, bc will be negative. Because both "a" and "c" are multiplied by "b" that is negative.

Finally, we have

**(a - b + c) ^{2} = a^{2} + b^{2} + c^{2} - 2ab - 2bc + 2ca**

**Case 3 :**

In (a + b + c)^{2}, if both "b" and "c" are negative, then we will have

(a - b - c)^{2}

How can we remember the expansion of (a - b - c)^{2} ?

It is very simple.

Let us consider the expansion of (a + b + c)^{2}_{.}

**(a + b + c) ^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca**

In the terms of the expansion above, consider the terms in which we find "b" and "c".

They are b^{2}, c^{2}, ab, bc, ac.

Even if we take negative sign for "b" in b^{2} and negative sign for "c" in c^{2}, the sign of both b^{2 }and c^{2} will be positive. Because they have even power 2.

The terms "ab" and "ca" will be negative.

Because, in "ab", "a" is multiplied by "b" that is negative.

Because, in "ca", "a" is multiplied by "c" that is negative.

The term "bc" will be positive.

Because, in "bc", both "b" and "c" are negative.

That is,

negative ⋅ negative = positive

Finally, we have

**(a - b - c) ^{2} = a^{2} + b^{2} + c^{2} - 2ab + 2bc - 2ca**

In the same way, we can get idea to remember the the expansions of

(a + b - c)^{3}, (a - b + c)^{3}, (a - b - c)^{3}

Apart from the stuff given above, if you would like to have problems on algebraic identities, please click the link given below.

**Worksheet on Algebraic Identities**

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