Algebraic Identities

Summary: Euler uses multiplication and squaring of compound quantities to derive reusable equalities: the binomial square, perfect square of any polynomial, difference of two squares, and sum/difference of two cubes.

Sources: chapter-1.2.3, chapter-1.2.6, chapter-1.2.10, additions-9

Last updated: 2026-05-10

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Binomial Square

From Chapter 1.2.6, squaring $a+b$ by direct multiplication gives (source: chapter-1.2.6)

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

The rule extends to any number of terms: the square of a polynomial equals the sum of the squares of all terms plus twice every cross-product. For three terms: (source: chapter-1.2.6)

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

Binomial Cube

From Chapter 1.2.9 (source: chapter-1.2.9):

$$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3.$$

The cube of a binomial contains the cube of each part plus $3ab(a+b)$.

Difference of Two Squares

Euler derives

$$(a+b)(a-b)=a^2-b^2.$$

(source: chapter-1.2.3)

He then observes that the difference of two square numbers is always composite, because it factors by the sum and difference of the roots. (source: chapter-1.2.3)

Sum and Difference of Two Cubes

From expanded products Euler obtains

$$(a+b)(a^2-ab+b^2)=a^3+b^3$$

and

$$(a-b)(a^2+ab+b^2)=a^3-b^3.$$

(source: chapter-1.2.3)

These identities become building blocks for larger factorizations, including one that yields $a^6-b^6$. (source: chapter-1.2.3)

Binomial Theorem (General Identity)

From Chapter 1.2.10 and 1.2.11, squaring and cubing are special cases of the general identity:

$$(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$$

where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$. The squares and cubes above correspond to $n=2$ and $n=3$ respectively. When $n$ is fractional or negative the identity produces an infinite-series. See binomial-theorem and pascal-triangle. (source: chapter-1.2.10)

Norm-Form Composition Identities (Add. IX)

Lagrange (additions-9, Arts. 89, 91) gives multiplicative identities for norm-forms of arbitrary degree.

Quadratic ($s^2-as+b=0$): $(x^2+axy+b{y}^2)(x^{\prime 2}+a{x}^{\prime }{y}^{\prime }+b{y}^{\prime 2})=X^2+aXY+b{Y}^2$ with $X=x{x}^{\prime }-by{y}^{\prime }$, $Y=x{y}^{\prime }+y{x}^{\prime }+ay{y}^{\prime }$. (Specialisation $a=0$ is the brahmagupta-fibonacci-identity.)

Squaring (set $x^{\prime }=x,y^{\prime }=y$): $(x^2+axy+b{y}^2{)}^2=(x^2-b{y}^2{)}^2+a(x^2-b{y}^2)(2xy+a{y}^2)+b(2xy+a{y}^2{)}^2.$

Cubing: $(x^2+axy+b{y}^2{)}^3=X^{\prime 2}+a{X}^{\prime }{Y}^{\prime }+b{Y}^{\prime 2}$ with $X^{\prime }=x^3-3bx{y}^2+ab{y}^3,Y^{\prime }=3x^{2}y+3ax{y}^2+(a^2-b)y^3.$

Cubic norm form ($s^3-a{s}^2+bs-c=0$): $N(x,y,z)=x^3+a{x}^{2}y+bx{y}^2+c{y}^3+(a^2-2b)x^{2}z+(ab-3c)xyz+(b^2-2ac)x{z}^2+ac{y}^{2}z+bcy{z}^2+c^2{z}^3$ satisfies $N(x,y,z)\cdot N(x^{\prime },y^{\prime },z^{\prime })=N(X,Y,Z)$ for explicit bilinear $X,Y,Z$. See composition-of-forms.

Role in the Book

These formulas show that multiplication of compound quantities is not only a computational rule but also a way to expose hidden structure in expressions. The binomial identities generalise into the binomial-theorem, one of the central results of the book. (source: chapter-1.2.3)

Related pages

• ch1.2.3-multiplication-compound-quantities
• ch1.2.6-squares-of-compound-quantities
• ch1.2.9-cubes-and-cube-root-extraction
• ch1.2.10-higher-powers-compound
• binomial-theorem
• pascal-triangle
• rationalization
• powers-and-exponents
• roots
• division-of-compound-quantities
• norm-forms
• composition-of-forms
• brahmagupta-fibonacci-identity
• add9-norm-forms-composition