Algebra: Arithmetic with Variables
Moving terms outside brackets
The distributive properties can also be read the other way around: \[a\cdot b+a\cdot c=a\cdot(b+c)\qquad\mathrm{and}\qquad a\cdot c+b\cdot c=(a+b)\cdot c\] This allows you to move a term outside the brackets and obtain a product of factors.
This works for numbers, but also for letters or for combinations of letters and numbers.
This example shows that there is more than one possibility for the factorized form: we could have also written \[-4a+12=-4(a-3)\] In general, we can move numbers outside of brackets. In this example, where all the numbers occurring are integers, we are not satisfied with \[-4a+12=2(-2a+6)\] because \[(-2a+6)=2(-a+3)\tiny.\] still holds. This gives us the chain \[-4a+12=2(-2a+6)=2\times 2(-a+3)=4(-a+3)\tiny.\] Because we focus on the expressions with variables, we are not interested in the factorization into prime factors of the constant #4#.
Factorization
The entire process of writing out an algebraic expression as a product is called factorizing. The result is called a factorization.
Except for the order of the terms in the product and the multiplication by constants the factorization is unique when there are no terms left to factorize.
It is important to indicate whether you are working with all the real numbers or only with the rational numbers. For example, #x^2-2# cannot be factorized for rational numbers, but it is with the real number #\sqrt{2}#; indeed, \[x^2-2=\left(x-\sqrt{2}\right)\cdot\left(x+\sqrt{2}\right)\tiny.\] This factorization will be discussed in more detail later.
Or visit omptest.org if jou are taking an OMPT exam.
