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: $$ a^2 + 2ab + b^2 = (a+b)^2 \\ a^2 - 2ab + b^2 = (a-b)^2 \\ a^2 - b^2 = (a+b)(a-b) \\ 1-a^{n}=(1-a)(1+a+a^{2}+ \cdots +a^{n-1}) $$
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)^2 $
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 2024-12-04,