Now on Google+! Add me to your circles!

I've finally gotten around to connecting my site to Google+!

I've set up a page specifically for this Math Concepts Explained blog.  Visit my Google+ page at https://plus.google.com/114879190180011260820 and add me to your circles!  I'm planning on posting brief bits of my blog posts there as I generate new ones, but you're going to have to visit this blog for the full deal!  You can also send me emails to ask questions or leave comments, if you don't want to do so on the blog's comment section.  Please send your math questions or comments to my new email address at mathconcepts101 [at] gmail [dot] com.

I'm also planning on starting a Math Concepts Explained profile on Facebook soon!  (Update:  My Facebook page is now live!  Come visit, click the Like button, and add me to your Friends lists!)

And for those of you who are new to my site, I also have a Twitter account already set up.  My profile is MathConcepts on Twitter (click the Follow me button near the top of the page).  You can easily connect with me on there as well!

Hopefully these new additions are going to be big for the blog!  Thanks to everyone for visiting!

Degree of a Polynomial

One of the most common questions you will find when first learning about polynomials, "How do you find the degree of a polynomial," sounds incredibly complicated and difficult, but really couldn't be simpler.  Sometimes, you don't need to use a bunch of math concepts to find the answer, or to do several mathematical manipulations to arrive at the solution.  Often, you don't have to do anything but just LOOK at the equation, and you can figure out the degree of a polynomial.  And that is because finding the degree of a polynomial is simple.

The degree of a polynomial is simply the highest exponent of the variable in the equation.

Look at this question, and you will see just how easy it can be:

What is the degree of the polynomial:  x³ + x = 5.

Yes, it is as simple as it sounds.  The highest power of x in this polynomial is 3, and so that is the degree.  See? Told you it can be easy!

Your questions will likely look similar to that one.  However, there are a few things to say as well, just in case you get a curve ball question.

Non-zero terms (e.g. integers) have a variable with an exponent of 0 (which means that the variable actually equals 1, and therefore you don't need to show it).  If you ONLY have non-zero integer, and therefore assume you have a variable with an exponent of 0, you can say that the degree of this non-zero constant polynomial is 0.  That is, constants are zero-degree.  However, that is only for non-zero polynomials.  Zero itself has an undefined degree.

So, the degree of the non-zero constant polynomial 8 is 0.  Easy.

Of course, you will inevitably get harder questions that do actually require a bit of work to arrive at your solution, but that will only be more of the same kind of mathematical concepts you've been studying, such as how to FOIL polynomials, or factoring by grouping.  In the end, when you have your polynomial expression, all you need to do is determine what is the highest exponent of the variable, and then state that as the degree.  Try some practice questions, and you will find that you won't be asking "how do you find the degree of a polynomial" anymore!

One last thought I'd like you to consider:  I have seen people searching my site for help with things like "degree polynomial" or "degree polynomials" and I think that sounds very strange, and suggests to me that this math concept is not being taught very well at all in the classroom.  Technically, you don't have "degree polynomials" per se.  ALL polynomials, except zero, have a degree which can be anything from third degree polynomials to higher degree polynomials such as ten-thousand degree polynomials (though I certainly don't want to be the one working with a polynomial of that degree!).  If you think about it, to say something like a "degree polynomial" is to be redundant!

In any case, if you arrived here after searching for help with finding the degree of a polynomial, or for "degree polynomial," I hope this post has been informative and helpful for you.  It's not a math concept that is as hard as it sounds, so hopefully my post makes sense and you can take something away to help teach other students who don't know what "degree polynomials" are.  :-)  As always, please remember to +1 me below if you liked my post!  Also follow me on Twitter with the button above!

Use Siri for Math on iPhone 4S

For people who recently bought Apple's new superphone, the iPhone 4S, one of the most compelling new features is the "virtual assistant" Siri.  It's a groundbreaking new app that does a pretty good job of understanding natural language spoken to it through the iPhone's mic, and then responding to you with some kind of acknowledgement or action.  Some of the cool things that Siri can do are highlighted everywhere in the news: send a text message to someone, call someone, setup an appointment or reminder.  In some countries (for now, only a few), Siri can even provide you with directions to a location, or understand things like "i'm hungry" and respond to you with nearby restaurants.  However, as useful as these functions are, one of Siri's less advertised features is an interaction with the knowledge engine Wolfram Alpha.

Wolfram Alpha works similarly to Google, but rather than returning a list of search results to a query, it actually attempts to answer your question with data in its gigantic fact database.  So, if you ask Wolfram Alpha "what is the fifth planet from the Sun" or "what is the capital city of Belgium," it will respond with "Jupiter" and "Brussels" as well as a boatload of information specifically about those entries.  Wolfram Alpha is truly amazing.  Now, if you put this together with Siri on the iPhone 4S, what you are able to do is simply ask Siri any factual question (without having to go into a browser or type out your question), it will query Wolfram Alpha for you, and then return the information you're looking for.

So, what does all of this have to do with math?  How does this allow you to use Siri for math on iPhone 4S?

Well, if you input math questions into Wolfram Alpha, it will come back with the answer.  This works especially well for unit conversions, or looking up constants, or working with math expressions.  Having the ability to simply ask Siri your math question, and have the app return the calculation for you is a remarkable shortcut to writing out and working through complex calculatons.  You don't even have to punch numbers into a calculator.  All you have to do is ask Siri your math questions, and it will return the solutions.

Of course this isn't going to TEACH you how to perform these calculations, but when you know about these math concepts already, this is an absolutely fantastic math shortcut that means you don't have to do the hard work!  Check out some of the ways that you can use Siri for math on iPhone 4S.  These are just a few examples, and I'm sure you can find even more.  (And remember that Wolfram Alpha can handle all kinds of factual questions, so you're not restricted to asking Siri only math questions!)  Once you realize how powerful Siri is, you will find that it can be a very powerful, helpful, and convenient calculator!

Show you the value of mathematical constants:

Calculate percentages:

Calculate square roots:

 Calculate reciprocals:

Perform unit conversions:

Perform arithmetic on a string of integers:

Perform arithmetic on fractions:

Perform complex calculations:

If you notice in my pictures, Siri doesn't just respond with the answer, but also with a host of other data such as unit conversions or different presentations of data.  It will also always show you what it understood for the original question, so you can verify that it understood you correctly and actually performed the computations on the proper input.

Now, admittedly, Siri is not going to help you learn math any easier or faster.  You still need to put in time with the usual techniques... i.e. you still have to do your homework and study!  And, you won't be allowed to take your iPhone 4S and Siri with you into an exam any time soon.  Just imagine everyone in an exam room talking to their iPhones and all the Siri's talking back!  However, when you get to a point in your studies that you understand the math concepts being taught, and you've done plenty of practice to understand how to arrive at the correct answer, then having the ability to use Siri for math on iPhone 4S to quickly and accurately find what you need to know is incredibly valuable!  (Please hit the Google +1 below if you found this article useful!)

How To Solve Equations

Knowing how to solve equations is a very important skill to have in mathematics courses.  There are all kinds of manipulations and substitutions that could be possible for any given equation, but knowing where and when to apply certain techniques is crucial to solving rational equations correctly.  In this post, I am going to go over several concepts that will be useful to you when it comes to solving rational equations.

To start, I will explain first-degree equations in one variable.  Quite simply, a first-degree equation is one in which there is only one variable.  In general, a first-degree or linear equation has the form:

ax + b = 0, where a and b are real numbers and a does not equal 0

I'm sure you are extremely familiar with this type of equation, though you may not know it by this name.  An example of a linear equation would be something like 3x - 12 = 0.  Undoubtedly, you can easily see that this equation is true when x = 4.  However, it is good to realize that the expression is neither true nor false until you substitute in a value for the variable.  Any value that makes the expression correct is called a solution or root of the equation.  To further classify this equation, 3x - 12 = 0 is also called a conditional equation, in that it is only true for certain values.

When you have two expressions that have the same solution (or root), these are called equivalent equations. Again, I'm sure you are familiar with the concept, but probably unfamiliar with this name.  When you have a first-degree equation in one variable, the general strategy that you typically employ is to express the equation equal to a series of equivalent equations, which you manipulate until you can reduce everything down to the solution to the equation.

The rules for generating equivalent equations are simple and intuitive.

1. You can add or subtract the same value from both sides of the equation.  (A corollary to this is that you can add AND subtract the same value on one side, without changing the other... since adding x and then subtracting x means you really have done nothing!)
2. You can multiply or divide each side of the equation by the same value.
3. You can simplify one side of the equation without affecting the other side of the equation.

I think these rules are fairly self-explanatory, so I'm not going to bother going into any examples to demonstrate them.

When you have arrived at your solution / root of your equation, it is ALWAYS smart to take that value and substitute it into the original expression to verify that it is indeed true.  It always amazes me how many people arrive at incorrect answers and leave it at that, when a simple review and check can either tell you that you are correct, or your answer needs more work.  ALWAYS REMEMBER TO CHECK YOUR ANSWERS!

By checking your answers by substituting the solution into the equation, you sometimes will determine that the solution you have found CANNOT be true, in which case your solution is called an extraneous root.  An example of this would be where, when checking your solution, you determine that you have a 0 on the bottom of a fraction (the denominator).  A fraction with a zero in the denominator is undefined, and so you can conclude that the root you determined does not satisfy your equation.  Extraneous roots may develop especially if you use rule number 2 above, but you multiply both sides by an EXPRESSION rather than a single number.  (eg. you multiply both sides by (x + 2))

That is all I am going to say about how to solve equations for now, especially the first-order equations (or linear equations).  I will continue in my next post with a discussion of solving quadratic equations.

Principal Axis Factoring

I'm just going to put a brief mention up here on principal axis factoring, a theoretical statistical method used to analyze common variance.  Principal axis factoring, also known as principal factor analysis or common factor analysis, is the most commonly used method in factor analysis.  According to Wikipedia, it "seeks the least number of factors which can account for the common variance (correlation) of a set of variables".  Factors are determined through the analysis of the common variance.


This concept of factor analysis is beyond the concepts that I would like to explain on my blog. It is much more of a university-level statistics concept than any of the other concepts that I have described so far.  If you would like a more thorough explanation of principal axis factoring, I highly suggest that you perform a Google search or find a good reference text book.  I mention it here only because of my recent posts on methods of factoring, and I was asked to specifically to reference this factoring method.