Polynomials in Mathematics

In this post, I would like to briefly refresh the concept of Polynomials in Mathematics, as at the start of the school year there are always students looking for extra polynomials help.

I would like to explain some basic terms and definitions that will hopefully make working with polynomials less confusing.

The first two terms which are fairly obvious are "variable" and "constant".  You probably don't need me to tell you that constants are terms that do not change (they are fixed... they don't always have to be known, but they do not change), whereas you can change the value of the variable to arrive at a different solution to your problem.  As in, if you vary the length of the square, you calculate a different area of the square.  The general practice is to assign letters near the start of the alphabet as constants (a, b, c), and at the end for variables (x, y, z).  These terms are self-explanatory, so I will leave them at that.

Substituting values into your variables only makes sense in some cases.  E.g.  sub in -4 for x, when the expression is square root of x.  It doesn't make sense.  Therefore, the set of numbers that DO make the expression make sense is called the "domain of the variable".

An example of this concept can be seen here.  Find the domain of the expression 1 / (x-5).

You can substitute any number in for x and calculate a solution, except for one.  You cannot have a zero as the denominator, as that makes the expression undefined.  So, in this case, you have to ask yourself "what makes the denominator equal zero?"  You can calculate in this case that x - 5 = 0, and so x=5.  Therefore, the domain of the original expression is all real numbers, except x cannot equal 5.

The term "polynomial" is given when you have a series of "terms" in an expression with different powers of x as long as the powers are not negative.  In general, a polynomial looks like this:

a_nxⁿ + a_n-1x^n-1 + ... a₁x + a₀

Each x value is contained in a "term".  The a values are "coefficients", which are constants.  They do not vary.  Overall, the "degree" of the polynomial is the highest power of x.  General convention is to right the terms from the highest degree of x to the lowest.

Since a polynomial is a generic collection of many (poly = many) terms, specific forms of polynomials have special names.  A "monomial" is a polynomial that only has one term, such as 2x.  A "binomial" has two terms, such as 4x² - 1x.  And a "trinomial" has three terms (guess what it looks like!).

Polynomial terms of the same degree can be combined.

In the expression 3x + 2x + 1, you can combine the terms so that you simplify to 5x + 1.  You can do this with x² terms and higher as well.

There are a couple of special polynomial products that will appear very frequently in your studies.  I will also go over these in my next post so that you can memorize them to apply them easily.  Hopefully, when you see these, you will understand how they work and will not need to look for polynomials help anymore!

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