Math Made Easy: Polynomial Characteristics

The entire purpose of math is to be a tool for us to use to explore, understand, and manipulate our universe. Without it, our current level of technology would be impossible. We use it to describe everything we create, and to predict how things are going to behave. To do this, we must understand just what those equations mean, what the physical reality behind them is, how they behave, and what altering one part of another of them exactly does. So, to that end, I’m going to address the behavior of polynomials, and just how we can easily determine what any given polynomial is going to do without even having to do much in the way of actual mathematics beyond just using our understanding of how equations behave.

So, just what *is* a polynomial? In short, a polynomial is any mathematical expression with more than one term, which expression is *not* written as a fraction. This second part is actually important. The reason being, is there’s a more technical definition of what a polynomial is that requires advanced mathematics – it’s all about a certain type of series, one that cannot be generated (created) with multiple terms in the denominator (the bottom of a fraction).

Oh, what is a term? A term is one part of an expression, separated from the other parts by either a plus sign or a minus sign. That’s kind of a clunky way to describe it, but it’s the simplest way to get the point across.

So, the most common type of polynomial people recognize is called the quadratic, and it looks like this:

x²+x+1

One term in that quadratic is the x².

The smallest polynomial we’ll ever see is a binomial, which only has two terms. How many terms can a polynomial have? Well, there’s no limit to that – we can, and do, have infinite polynomials, and we can even evaluate those with some clever thinking and calculus. But, that’s not the point of this article – I’m going to address finite ones, and how you can quickly determine their behavior by looking at them. The most important term for determining the behavior of the polynomial is that term which has the largest degree. The degree of a term is the value of the exponent. So, in a quadratic, the degree is two, since the largest term is the one with the two as an exponent. There are four things we can determine from this term.

The first, is: is this guy a cupped curve, called a parabola, perhaps with some squiggly bumps in it, or is it more of a line (but still a curve), also perhaps with some squiggly bumps in it. Again, we look at the degree. If it is even, it is a cupped curve, and will have some bumps in it if it is any degree larger than 2. If the degree is odd, then it is a curved line (straight line for degree of one, curved with bumps, or turns, for any degree larger).

Second, we can determine whether it opens up or down (for even degreed parabolas) or goes up and down (for the odd degreed lines). If the term with the largest exponent is positive, then it is an “up” polynomial – the cup opens up or the line goes up as it moves from left to right on the graph. Polynomials with that term being negative, we call “down” as the parabola opens down, or the line goes down as it moves from right to left on the graph.

The third thing we can tell from this largest degreed term (remember, the one with the largest exponent), is how quickly it goes up and down, or how steep it is (the slope). This is determined by the coefficient of the term – the number that is multiplying the variable. If this number is a whole number, then the bigger it is, the steeper the slope, or the faster the figure goes up or down. If that number is a fraction, then the slope becomes shallower the smaller the fraction is, spreading out the parabola, or making the line less and less steep (whichever the figure is).

Finally, the number of turns the curve takes depends solely on the largest degree. Subtract one from it, and you’ve determine the number of curves. For instance, a basic parabola, with no squiggles, is a quadratic with a degree of 2. Two minus one is one, and, indeed, the basic parabola only makes one turn. A cubic has a degree of 3, and thus makes two turns, which is the cause of it being essentially a line with a squiggle in it rather than a cup.

There are two other basic physical behaviours we can determine from the polynomial: horizontal and vertical shift, and those are obtain from either the constant in the polynomial or, you guessed it, the term of highest degree. The constant shifts the figure up and down – a negative constant shifts it down. A positive constant shifts it down. For horizontal shift, we look to see if something is added to or subtracted from the highest degree term before the power is applied (ie, (x+1)²). If something is added, the figure is shifted to the left. If something is subtracted, it is shifted to the right.

And that’s it. Six basic physical behaviours we can see in all polynomials without really having to do much in the way of computation. We are able to learn their end behavior (cup or line with squiggles), whether they go up or down, how steep they are, their number of turns, and whether there is a horizontal or vertical shift. Understanding this helps us to visualize the expression and get a better feeling for what it means.