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Math Expression Factorization

Tool to factorize a math expression. Factorization of a mathematical expression consists in expressing it as a product, it is the inverse of an expansion.

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Math Expression Factorization -

Tag(s) : Symbolic Computation

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# Math Expression Factorization

## Expression Factorization

### Automatic factorization

 Factorisation Classic (Recommended for polynomials) Classic (complex allowed) Full (for polynomial with roots) Trigonometric (cos, sin, etc.)

## Prime Numbers Factorization

### What is factorization? (Definition)

Factorizing is the action of transforming a sum (an addition) into a product (a multiplication) of 2 factors (or more).

Example: The addition $3 x + 6$ can be factorized as the multiplication $3 \times (x + 2)$

Factorization is a mathematical transformation modifying the writing of an expression without changing the result.

Factorization is the inverse transformation of expansion which consists of transforming a product into a sum

### How to factorize a polynomial-like expression?

Several methods of factorization exist in mathematics:

Factorization by finding a common factor

Example: The addition $3a + 3b$ has two terms ($3a$ and $3b$) that have the common factor $3$, so $3a+3b = 3(a+b)$

Factorization by identifying a remarkable identity

The most common outstanding identities allowing factorization are: $$a^2 + 2ab + b^2 = (a+b)^2 \\ a^2 - 2ab + b^2 = (a-b)^2 \\ a^2 - b^2 = (a+b)(a-b)$$

Example: The expression $x^2+2x+1$ contains a remarkable identity of the form $a^2 + 2ab + b^2$ (with $a = x$ and $b = 1$) so it can be factorized $x^2+2x+1 = (x+1)(x-1) \\ 1-a^{n}=(1-a)(1+a+a^{2}+...+a^{n-1})$

Factorization with polynomial roots

By knowing (or calculating) all the roots $\alpha_i$ of a polynomial of variable $x$, then this one can be factorized as the product of the $(x-\alpha_i)$

Example: The polynomial $x^2 - 2$ has roots $x = \sqrt{2}$ and $x = -\sqrt{2}$ so it is factorized $(x-\sqrt{2})(x+\sqrt{2})$

### What is integer factorization?

Factorization can also be applied to whole numbers, in order to determine if they are multiples of other numbers.

Example: $8$ can be factorized $2 \times 4$ or $4 \times 2$ or $2 \times 2 \times 2$

If an integer has no factors other than 1 and itself then it is a prime number.

The process of factoring an integer is also called prime number decomposition.

### How to factorize a trigonometric expression?

dCode factorizes trigonometric expression in order to simplify them by expressing them with sin and cos

Example: $$1+1/\sec(x) = 2\cos(x/2)^2$$

Example: $$\cos(x+y) + \sin(x)\sin(y) = \cos(x)\cos(y)$$

### How to display steps by steps?

The solver/factorizer has no real steps, at least not steps similar to those required of college or high school. For the moment steps are not displayed, but the solver allows checking a result.

## Source code

dCode retains ownership of the "Math Expression Factorization" source code. Except explicit open source licence (indicated Creative Commons / free), the "Math Expression Factorization" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Math Expression Factorization" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Math Expression Factorization" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
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Math Expression Factorization on dCode.fr [online website], retrieved on 2022-12-09, https://www.dcode.fr/math-expression-factor

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