# factor

Factorization

## Description

uses additional options specified by one or more `F`

= factor(___,`Name,Value`

)`Name,Value`

pair
arguments. This syntax can use any of the input arguments from the
previous syntaxes.

## Examples

### Factor Integer Numbers

F = factor(823429252)

F = 2 2 59 283 12329

To factor integers greater than `flintmax`

,
convert the integer to a symbolic object using `sym`

.
Then place the number in quotation marks to represent it accurately.

F = factor(sym('82342925225632328'))

F = [ 2, 2, 2, 251, 401, 18311, 5584781]

To factor a negative integer, convert it to a symbolic object
using `sym`

.

F = factor(sym(-92465))

F = [ -1, 5, 18493]

### Perform Prime Factorization of Large Numbers

Perform prime factorization for `41758540882408627201`

.
Since the integer is greater than `flintmax`

, convert
it to a symbolic object using `sym`

, and place
the number in quotation marks to represent it accurately.

n = sym('41758540882408627201'); factor(n)

ans = [ 479001599, 87178291199]

### Factor Symbolic Fractions

Factor the fraction `112/81`

by
converting it into a symbolic object using `sym`

.

F = factor(sym(112/81))

F = [ 2, 2, 2, 2, 7, 1/3, 1/3, 1/3, 1/3]

### Factor Polynomials

Factor the polynomial `x^6-1`

.

```
syms x
F = factor(x^6-1)
```

F = [ x - 1, x + 1, x^2 + x + 1, x^2 - x + 1]

Factor the polynomial `y^6-x^6`

.

```
syms y
F = factor(y^6-x^6)
```

F = [ -1, x - y, x + y, x^2 + x*y + y^2, x^2 - x*y + y^2]

### Separate Factors Containing Specified Variables

Factor `y^2*x^2`

for factors
containing `x`

.

syms x y F = factor(y^2*x^2,x)

F = [ y^2, x, x]

`factor`

combines all factors without `x`

into
the first element. The remaining elements of `F`

contain
irreducible factors that contain `x`

.

Factor the polynomial `y`

for factors containing
symbolic variables `b`

and `c`

.

syms a b c d y = -a*b^5*c*d*(a^2 - 1)*(a*d - b*c); F = factor(y,[b c])

F = [ -a*d*(a - 1)*(a + 1), b, b, b, b, b, c, a*d - b*c]

`factor`

combines all factors without `b`

or `c`

into
the first element of `F`

. The remaining elements
of `F`

contain irreducible factors of `y`

that
contain either `b`

or `c`

.

### Choose Factorization Modes

Use the `FactorMode`

argument
to choose a particular factorization mode.

Factor an expression without specifying the factorization mode.
By default, `factor`

uses factorization over rational
numbers. In this mode, `factor`

keeps rational
numbers in their exact symbolic form.

syms x factor(x^3 + 2, x)

ans = x^3 + 2

Factor the same expression, but this time use numeric factorization over real numbers. This mode factors the expression into linear and quadratic irreducible polynomials with real coefficients and converts all numeric values to floating-point numbers.

factor(x^3 + 2, x, 'FactorMode', 'real')

ans = [ x + 1.2599210498948731647672106072782,... x^2 - 1.2599210498948731647672106072782*x + 1.5874010519681994747517056392723]

Factor this expression using factorization over complex numbers.
In this mode, `factor`

reduces quadratic polynomials
to linear expressions with complex coefficients. This mode converts
all numeric values to floating-point numbers.

factor(x^3 + 2, x, 'FactorMode', 'complex')

ans = [ x + 1.2599210498948731647672106072782,... x - 0.62996052494743658238360530363911 + 1.0911236359717214035600726141898i,... x - 0.62996052494743658238360530363911 - 1.0911236359717214035600726141898i]

Factor this expression using the full factorization mode. This mode factors the expression into linear expressions, reducing quadratic polynomials to linear expressions with complex coefficients. This mode keeps rational numbers in their exact symbolic form.

factor(x^3 + 2, x, 'FactorMode', 'full')

ans = [ x + 2^(1/3),... x - 2^(1/3)*((3^(1/2)*1i)/2 + 1/2),... x + 2^(1/3)*((3^(1/2)*1i)/2 - 1/2)]

Approximate the result with floating-point numbers by using `vpa`

.
Because the expression does not contain any symbolic parameters besides
the variable `x`

, the result is the same as in complex
factorization mode.

vpa(ans)

ans = [ x + 1.2599210498948731647672106072782,... x - 0.62996052494743658238360530363911 - 1.0911236359717214035600726141898i,... x - 0.62996052494743658238360530363911 + 1.0911236359717214035600726141898i]

### Approximate Results Containing `RootOf`

In the full factorization mode,`factor`

also
can return results as a symbolic sums over polynomial roots expressed
as `RootOf`

.

Factor this expression.

syms x s = factor(x^3 + x - 3, x, 'FactorMode','full')

s = [ x - root(z^3 + z - 3, z, 1),... x - root(z^3 + z - 3, z, 2),... x - root(z^3 + z - 3, z, 3)]

Approximate the result with floating-point numbers by using `vpa`

.

vpa(s)

ans = [ x - 1.2134116627622296341321313773815,... x + 0.60670583138111481706606568869074 + 1.450612249188441526515442203395i,... x + 0.60670583138111481706606568869074 - 1.450612249188441526515442203395i]

## Input Arguments

## Output Arguments

## Tips

To factor an integer greater than

`flintmax`

, wrap the integer with`sym`

. Then place the integer in quotation marks to represent it accurately, for example,`sym('465971235659856452')`

.To factor a negative integer, wrap the integer with

`sym`

, for example,`sym(-3)`

.

## Version History

**Introduced before R2006a**