**Algebraic Identities :**

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 algebra 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.

(1) (a + b)² = a² + 2 ab + b² (2) (a + b)² = (a - b)² + 4 ab | |

(3) (a - b)² = a² - 2 ab + b² (4) (a - b)² = (a + b)² - 4 ab | |

(5) a² - b² = (a + b) (a - b) | |

(6) (x+a) (x+b) =x² + (a + b) x+a b | |

(7) (a+b)³=a³+3a²b+3ab²+b³ (8) (a+b)³=a³+b³+3ab(a+b) | |

(9) (a-b)³=a³-3a²b+3ab²-b³ (10) (a-b)³=a³-b³-3ab(a-b) | |

(11) a³+b³ = (a+b)(a²-ab+b²) (12) a³+b³=(a+b)³-3 ab(a + b) | |

(13) a³-b³= (a-b)(a²+ab+ b²) (14) a³-b³=(a-b)³ +3ab(a-b) |

(15) (a+b+c)²= a²+b²+c² +2ab+2bc+2ca (16) (a+b-c)²=a²+b²+c² +2ab-2bc-2ca (17) (a-b+c)²= a²+b²+c²-2ab-2bc+2ca (18) (a-b-c)²= a²+b²+c²-2ab+2bc-2ca (19) a² + b² = (a + b)² - 2ab (20) a² + b² = (a - b)² + 2ab (21) a² + b²=½ [(a+b)²-(a-b)²] (22) ab = ¼[(a+b)²- (a - b)²] |

(23) (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3b²c + 3ac² + 3bc² + 6abc

(24) (a + b - c)³ = a³ + b³ - c³ + 3a²b - 3a²c + 3ab² - 3b²c + 3ac² + 3bc² - 6abc

(25) (a - b + c)³ = a³ - b³ + c³ - 3a²b + 3a²c + 3ab² + 3b²c + 3ac² - 3bc² - 6abc

(26) (a - b - c)³ = a³ - b³ - c³ - 3a²b - 3a²c + 3ab² - 3b²c + 3ac² - 3bc² + 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**

After having gone through the stuff given above, we hope that the students would have understood algebraic Identities.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**