Algebra

Equations

Inequalities

Graphs

Numbers

Calculus

Matrices


Algebra : Expand

Basic | Intermediate | Advanced | Help

Description

The expand command will multiply out products and positive integer powers in an algebraic expression. Its main use is to convert polynomials to a form containing a simple sum of terms only.

To use the expand command, simply go to the basic expand page, type in your expression and hit the "Expand" 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 expand page, where there are a number of options which allow you to customize the command.

Examples

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

Basic

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

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

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

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

Advanced

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

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

Expression : (1 + x)^2 + (1 + y)^2
Options : Parts matching 1 + x
Result : 1 + 2 x + x^2 + (1 + y)^2

Expression : (1 + x)^2/(1 + y)^2
Options : Scope - default
Result : (1 + y)^(-2) + (2 x)/(1 + y)^2 + x^2/(1 + y)^2

Expression : (1 + x)^2/(1 + y)^2
Options : Scope - all
Result : (1 + 2 y + y^2)^(-1) + (2 x)/(1 + 2 y + y^2) + x^2/(1 + 2 y + y^2)

Expression : (1 + x)^2/(1 + y)^2
Options : Scope - numerator
Result : (1 + 2 x + x^2)/(1 + y)^2

Expression : (1 + x)^2/(1 + y)^2
Options : Scope - denominator
Result : (1 + x)^2/(1 + 2 y + y^2)

Options (advanced page only)

Trig functions

Values : checked or unchecked
Default : unchecked

When Trig functions is checked, the Expand command will carry out trigonometric as well as algebraic transformations. In particular, trigonometric functions of sums and differences will be expanded in full.

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


Modulo

Values : checked or unchecked + empty string or positive integer or zero
Default : unchecked + empty string

When Modulo is checked and a positive integer is entered into the text field, the result of the expansion is reduced modulo n. When Modulo is unchecked or the text field contains 0 or is empty, the expansion is carried out over the field of integers.

For example, if the modulo is set to 3, then (1 + a)3 will be expanded as 1 + a3 rather than 1 + 3 a + 3 a2 + a3.


Parts matching

Values : checked or unchecked + empty string or expression
Default : unchecked + empty string

When Parts matching is checked and an expression has been entered into the text field, the Expand command will only expand out those parts of the expression matching the given pattern.

For example, if the pattern is set to 1 + a, then (1 + a)2 + (1 + b)3 will be expanded as (1 + 2 a + a2) + (1 + b)3.


Scope

Values : default or all or numerator or denominator
Default : default

The Scope option determines which parts of an expression are expanded. The Default setting expands the numerator of each term in an expression and splits fractions into several parts if possible, canceling common factors as necessary. The Numerator and Denominator settings expand only the numerators and denominators of terms respectively, whilst the All setting does all possible expansions in the numerators and denominators of each term.