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.

Math Expression Factorization - dCode

Tag(s) : Symbolic Computation

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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

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}) $

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.

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

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.

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*Math Expression Factorization* on dCode.fr [online website], retrieved on 2022-12-09,

factorisation,factor,factorize,polynomial,remarkable,identity,expression,math,product,sum,var

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