Wolfram|Alpha is written in Mathematica, which as its name suggests is a fantastic system for doing mathematics. Strong algorithms for algebraic simplification have always been a central feature of computer algebra systems, so it should come as no surprise to know that Mathematica excels at simplifying algebraic expressions. The main two commands for simplifying an expression in Mathematica are Simplify and FullSimplify. There are also many specific commands for expressing an algebraic expression in some form. For example, if you want to expand a product of linear polynomials, Expand is the appropriate function.
The good news is that everyone has access to the power of Mathematica‘s simplification and algebraic manipulation commands in Wolfram|Alpha. We will now outline some of these features in Wolfram|Alpha, starting with the expression:
and we will use Wolfram|Alpha to break it down to something significantly much simpler.
The expression simplifies to zero (excluding |x| = 3). Let’s take a closer look at this simplification. First, the denominators. We will expand the polynomial (x – 3)(x + 3) and see that it is equal to the other denominator.
So we have a common denominator:
Now we can simplify the numerator:
We see that it is zero.
Let’s look at some other examples. We can express sin(n x) in terms of a polynomial in sin(x) and cos(x) by asking Wolfram|Alpha:
We can factor polynomials over their splitting fields:
Form a single fraction from a number of terms:
Express a single fraction in partial fraction form:
And solve equations:
We now have implemented a new simplification program for Wolfram|Alpha, which allows Wolfram|Alpha to find even more alternate and simplified forms for algebraic expressions. Here are a couple of examples.
In the near future, we will have the functionality to show the steps used to derive an algebraic simplification. Stay tuned to this blog for more details!