List of Algebra Formulas

Algebra is one of the most important branches of mathematics. Below is a complete list of algebra formulas including basic identities, advanced formulas, and their alternative forms.

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1. Basic Algebraic Identities

$$\begin{array}{l}\left( i \right){\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\\\quad \quad \quad \quad \; = {\left( {a - b} \right)^2} + 4ab\\\left( {ii} \right){\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\\\quad \quad \quad \quad \; = {\left( {a + b} \right)^2} - 4ab\\\left( {iii} \right){a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\\\quad \quad \quad \quad \;\; = {\left( {a - b} \right)^2} + 2ab\\\quad \quad \quad \quad \;\; = \frac{1}{2}\left\{ {{{\left( {a + b} \right)}^2} + {{\left( {a - b} \right)}^2}} \right\}\\\left( {iv} \right)2\left( {{a^2} + {b^2}} \right) = {\left( {a + b} \right)^2} + {\left( {a - b} \right)^2}\\\left( v \right){a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\\\left( {vi} \right)4ab = {\left( {a + b} \right)^2} - {\left( {a - b} \right)^2}\\\left( {vii} \right)ab = {\left( {\frac{{a + b}}{2}} \right)^2} - {\left( {\frac{{a - b}}{2}} \right)^2}\end{array} $$

2. Cube Identities

$$\begin{array}{l}\left( i \right){\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}\\\quad \quad \quad \quad \; = {a^3} + {b^3} + 3ab\left( {a + b} \right)\\\left( {ii} \right){\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}\\\quad \quad \quad \quad \; = {a^3} - {b^3} - 3ab\left( {a - b} \right)\\\left( {iii} \right){a^3} + {b^3} = {\left( {a + b} \right)^3} - 3ab\left( {a + b} \right)\\\quad \quad \quad \quad \;\; = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\\\left( {iv} \right){a^3} - {b^3} = {\left( {a - b} \right)^3} + 3ab\left( {a - b} \right)\\\quad \quad \quad \quad \;\; = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\end{array} $$

3. Special Algebraic Identities

$$\begin{array}{l}\left( i \right){(x + y + z)^2} = {x^2} + {y^2} + {z^2} + 2(xy + yz + zx)\\\left( {ii} \right){x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)\\\quad \quad \quad \quad \quad \quad \quad \quad \;\;\; = \frac{1}{2}\left[ {\left( {x + y + z} \right)\left\{ {{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}} \right\}} \right]\\\left( {iii} \right)\;{\text{If }}x + y + z = 0{\text{, then }}{x^3} + {y^3} + {z^3} = 3xyz\end{array} $$

This collection of algebra formulas and their alternative forms is extremely useful for students. Memorizing these will save time in exams and make problem solving much faster.