Algebra : Factor

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The factor command essentially does the reverse of the expand command, attempting to write sums of terms as products of factors. It knows all the rules of factorization, including common factors, differences of two squares, sums and differences of cubes, quadratics and a whole lot more.

To use the factor command, simply go to the basic factor page, type in your expression and hit the "Factor" button. Your question will be automatically answered by computer and the reply will be shown in your browser within a few seconds. If you would like more control over how your question is answered, try the advanced factor page, where there are a number of options which allow you to customize the command.


Here are some examples illustrating the types of expressions you can use the factor command on and the results which QuickMath will return.


Expression : 1 + 2 x + x^2
Result : (1 + x)^2

Expression : -27 + 54 x - 36 x^2 + 8 x^3
Result : (2 x - 3)^3

Expression : a^8 + 4 a^6 b^3 + 6 a^4 b^6 + 4 a^2 b^9 + b^12
Result : (a^2 + b^3)^4

Expression : 1 - 2 e^x + e^(2 x) + 2 sin(x) - 2 e^x sin(x) + sin(x)^2
Result : (1 + sin(x) - e^x)^2


Expression : cos(b) sin(a) + cos(a) sin(b)
Options : Trig functions
Result : sin(a + b)

Expression : a^10 + 2 a^5 + 1
Options : Modulo 5
Result : (1 + a)^10

Expression : 5 - 5 x - 5 x^2 + 5 x^3
Options : List factors
Result : {{5, 1}, {-1 + x, 2}, {1 + x, 1}}

Expression : x^2 + 2 sqrt(2) x + 2
Options : Field extensions - automatic
Result : (sqrt(2) + x)^2

Expression : a^4 + b^4
Options : Field extensions - custom : sqrt(2)
Result : -((-a^2 + sqrt(2) a b - b^2) (a^2 + sqrt(2) a b + b^2))

Options (advanced page only)

Trig functions

Values : checked or unchecked
Default : unchecked

When Trig functions is checked, the Factor command will carry out trigonometric as well as algebraic transformations. In particular, trigonometric products will be factored in full.

For example, cos(b)sin(a) + cos(a)sin(b) would be factored as sin(a+b).


Values : checked or unchecked + empty string or prime number or zero
Default : unchecked + empty string

When Modulo is checked and a prime number is entered into the modulo field, the factorization is carried out modulo n. When Modulo is unchecked or the text field contains 0 or is empty, the factorization is carried out over the field of rational numbers.

For example, if the modulo is set to 3, then 1 + a3 will be factored as (1 + a)3, whereas by default, 1 + 3 a + 3 a2 + a3 would be factored as (1 + a)3.

Gaussian integers

Values : checked or unchecked
Default : unchecked

The Gaussian integers option can be used to factor expressions over the field of Gaussian integers. This is the set of complex numbers of the form a + i b, where a and b are integers. Allowing factoring to take place over this field considerably extends the scope of the Factor command.

For example, over the field of rationals, 1 + x2 cannot be factored. However, over the field of Gaussian integers, it can be factored as (1 + i x) (1 - i x).

List factors

Values : checked or unchecked
Default : unchecked

When the List factors option is unchecked, the Factor command simply tries to return a product of factors which is equivalent to the expression submitted. When this option is checked, however, the factors are returned in a list. Each factor is paired with the exponent that it is raised to.

For example, if List factors is unchecked, then sending 5 + 70 x + 380 x2 + 1000 x3 + 1280 x4 + 640 x5 to be factored will result in the answer 5 (1 + 2 x)3 (1 + 4 x)2. However, if List factors is checked, then the answer will be {{5, 1}, {1 + 2 x, 3}, {1 + 4 x, 2}}. This indicates that the factors are 51, (1 + 2 x)3 and (1 + 4 x)2.

Field extensions

Values : none or automatic or custom + custom extension expressions (only if the custom option is selected)
Default : none

Choosing the none option means that factoring will be carried out over the field of rationals (default) ot the field of Gaussian integers, if this option has been previously selected.

Choosing the automatic option will extend the field over which factorization is being attempted by any algebraic numbers appearing in the expression. For example, the expression 2 + sqrt(2) x + sqrt(2) y + x y cannot be factored over the field of rational numbers. However, if the automatic option is checked, the answer can now include sqrt(2), and it is (sqrt(2) + x) (sqrt(2) + y).

Choosing the custom option enables you to choose your own field extensions.


Values : complete or square free or terms + terms to be isolated (only if the terms option is selected)
Default : complete

Choosing the complete option means that factoring will be carried out as completely as possible over the specified field (the rationals by default).

Choosing the square free option will pull out any multiple factors in a polynomial.

Choosing the terms option whilst leaving the terms text field empty will pull out any overall numerical factor in the expression. Entering one or more terms in the terms text area will cause the factor command to pull out overall factors in the expression which do not depend on any of the specified terms.