Expand

Expand out products and positive integer powers; for a hypercomplex operand, puts $x$ in blade normal form — the same result the multivector arithmetic already produces.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageExpand✓↗

Expand(x)expand out products and positive integer powers.

SageMathSR(x).expand()SymPyexpand(x)

Typed signatures

Type	Package	Applies
(value) -> value	combinatoricsreplaces hypercomplex	always
(value) -> value	compute-engine	always
(value) -> value	hypercomplexreplaces compute-engine	always

Domain: Hypercomplex algebra

Examples

Basic #

Expand $(x+1)^2$ into $x^2 + 2x + 1$

live ↗

In$\mathrm{Expand}((x+1)^2)$Out$x^2+2x+1$

Expand $(x+1)(x-1)$ into the difference of squares

live ↗

In$\mathrm{Expand}((x+1)(x-1))$Out$x^2-1$

Multiplies out a product of two binomials

live ↗

In$\mathrm{Expand}((x+3)(x+2))$Out$x^2+5x+6$

A product of three polynomial factors collapses to one degree-10 polynomial

live ↗

In$\mathrm{Expand}((1+x)^3(5x^2+1)(x^5+1))$Out$5x^{10}+15x^9+16x^8+8x^7+3x^6+6x^5+15x^4+16x^3+8x^2+3x+1$

The pure fifth powers cancel, leaving only mixed terms

live ↗

In$\mathrm{Expand}((x+y)^5-(x^5+y^5))$Out$5yx^4+5xy^4+10y^3x^2+10x^3y^2$

Scope #

Expand $(x+y)^3$, a two-variable binomial power

live ↗

In$\mathrm{Expand}((x+y)^3)$Out$x^3+y^3+3yx^2+3xy^2$

A product of conjugates squared: $(x^2-y^2)^2$ expanded

live ↗

In$\mathrm{Expand}((x+y)^2(x-y)^2)$Out$x^4+y^4-2x^2y^2$

A negative power in the denominator is kept, only the numerator is expanded

live ↗

In$\mathrm{Expand}(\frac{(x+y)^2}{(x-y)^2})$Out$\frac{x^2}{(x-y)^2}+\frac{y^2}{(x-y)^2}+\frac{2xy}{(x-y)^2}$

Expands around non-polynomial atoms like $\sin x$

live ↗

In$\mathrm{Expand}((\sin(x)+\cos(x))^2)$Out$\sin(x)^2+\cos(x)^2+2\sin(x)\cos(x)$

Under a root the base is not expanded, only reordered

live ↗

In$\mathrm{Expand}(\sqrt{(1+x)^2})$Out$\sqrt{(x+1)^2}$

Same shape with a different denominator: terms are split over it

live ↗

In$\mathrm{Expand}(\frac{(x+y)^2}{(z+y)^2})$Out$\frac{x^2}{(y+z)^2}+\frac{y^2}{(y+z)^2}+\frac{2xy}{(y+z)^2}$

The binomial theorem row for $n=5$ as an identity

live ↗

In$x^5+5x^4+10x^3+10x^2+5x+1=\mathrm{Expand}((x+1)^5)$Out$\mathrm{True}$

live ↗

In$\mathrm{Expand}((x+i)(x-i))$Out$x^2+1$

live ↗

In$\mathrm{Expand}((x^{s}+2e^{-x})^3)$Out$x^{3s}+6e^{-x}x^{2s}+12e^{-2x}x^{s}+8e^{-3x}$

Applications #

Four factors collapse to $(x^2-1)^2$

live ↗

In$\mathrm{Expand}((1-x)(-x-1)(1-x)(-x-1))$Out$x^4-2x^2+1$

Properties #

Multiple variables expand and collect by powers of $x$

live ↗

In$\mathrm{Expand}((1+x+y)(2-x)^3)$Out$-x^4-yx^3+5x^3+6yx^2-6x^2-12xy-4x+8y+8$

Coefficients of $(x+1)(x+2)(x+3)$ are Stirling numbers of the first kind

live ↗

In$\mathrm{Expand}((1+x)(2+x)(3+x))$Out$x^3+6x^2+11x+6$

live ↗

In$\mathrm{Expand}(\prod_{i=6}(x+i))$Out$x^6+21x^5+175x^4+735x^3+1\,624x^2+1\,764x+720$

live ↗

In$\mathrm{Expand}(\prod_{k=6}(x+k))$Out$x^6+21x^5+175x^4+735x^3+1\,624x^2+1\,764x+720$

About

Examples	19
Checked against	Wolfram Language 19 agreeSageMath 17 agree1 differSymPy 18 agree1 differ
Crosswalk	Wolfram LanguageSageMathSymPy

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

See also: NonCommutativeMultiply, Basis