List of algebraic identities

File:Binomio al cubo.svg

In mathematics, algebraic identities are equalities that involve algebraic functions and are true for every value of the occurring variables where both sides of the equality are defined.

These identities are useful whenever algebraic expressions need to be simplified. An important application is the integration of algebraic functions.

In middle school in the Algebra course, some common identities (like Difference of two squares) are studied, mainly the cases of multiplication of polynomials. Many of them are a special case of the binomial theorem.

Second degree formulas

In the image, decomposition of a square of side a + b, into one square of side a, one square of side b and two rectangles of sides a and b.

• $(a\pm b{)}^2=a^2\pm 2ab+b^2$
• $a^2-b^2=(a+b)(a-b)$ (see Difference of two squares)
• ${(a+b+c)}^2=a^2+b^2+c^2+2ab+2ac+2bc$

Third degree formulas

• $(a\pm b{)}^3=a^3\pm 3a^{2}b+3a{b}^2\pm b^3$
• $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$
• ${(a+b+c)}^3=a^3+b^3+c^3+3a^{2}b+3a^{2}c+3a{b}^2+3a{c}^2+3b^{2}c+3b{c}^2+6abc$

Fourth degree formulas

• $(a\pm b{)}^4=a^4\pm 4a^{3}b+6a^2{b}^2\pm 4a{b}^3+b^4$
• $a^4-b^4=(a-b)(a+b)(a^2+b^2)$ (derived from $a^2-b^2$)

Formulas for the nth degree

For any natural number n, the following formulas hold.

• $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}{b}^2+\dots +a^2{b}^{n-3}+a{b}^{n-2}+b^{n-1})$
• $a^{2n}-b^{2n}=(a+b)(a^{2n-1}-a^{2n-2}b+a^{2n-3}{b}^2-\dots -a^2{b}^{2n-3}+a{b}^{2n-2}-b^{2n-1})$
• $a^{2n}-b^{2n}=(a^n+b^n)(a^n-b^n)$
• $a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}{b}^2-\dots +a^2{b}^{2n-2}-a{b}^{2n-1}+b^{2n})$

Some properties of formulas

• $(a-b{)}^{2n}=(b-a{)}^{2n}$, where $n\in N$
• $(a-b{)}^{2n+1}=-(b-a{)}^{2n+1}$, where $n\in N$

See also

• Binomial theorem
• Factorization of polynomials

Sources

• Dickson, Leonard Eugene (2005) [1920], History of the theory of numbers. Vol. II: Diophantine analysis, New York: Dover Publications, ISBN 978-0-486-44233-4, MR 0245500, Zbl 1214.11002

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