It's been a while since we mentioned them, and now your brain is in a total "exponent" zone. It's okay. We're here for you.
Like aliens, polynomials come in many shapes, sizes, and extradimensional configurations. Just our theory. Anyway, the diversity of polynomials is the reason you're dissecting them. It's important to realize exactly what kind of polynomial you have in your hands. To continue with the alien analogy, there's a big difference between E.T. and that baddie from Cloverfield. It's good to know which type you're dealing with before making arrangements to meet them in the middle of the woods for a picnic.
Like we said earlier, a polynomial is an expression containing constants and variables that can be combined using addition, subtraction, and multiplication. Here's the kicker: the exponents on all variables must be positive whole numbers. Otherwise, it ain't a polynomial.
We determine what kind of polynomial we have by looking at its parts, or terms. Terms like 2x tend to look relatively simple, though they also have their own parts with fancy shmancy names like coefficient (the number attached to the term by multiplication) and variable (a letter representing an unknown value). All the coefficients and constants in a polynomial need to be real numbers. Terms also have exponents—always.
If a term appears not to have an exponent, that means its exponent is 1. It's still there, but you can't see it. Like one of those stars in the night sky that disappear when you try to look at it directly. Come on, star. Work with us here.
Let's look at a few polynomials.
The expression is a polynomial.
Where's the x in the last term? Hiding, of course. Seriously, it needs to get over its debilitating shyness. Since x = 1 (remember that anything raised to the power of 0 is 1), we can think of that last term as 4x0, which is a real number multiplied by a whole number power of x.
The expression 4x2 – 3x – 1 is also polynomial.
The first term of this polynomial is 4x2. Since we could rewrite the polynomial as 4x2 + (-3)x + (-1), the second term of this polynomial is -3x and the third term is -1. Watch those negative signs. They'll getcha.
In fact, the following are all polynomials.
It's important to remember that not everything that looks like a polynomial is one. Some are wolves in polynomial's clothing. Not all terms, coefficients, and exponents add up to an authentic polynomial. For instance, all our exponents in polynomials must be positive whole numbers. Them's the rules.
The following are NOT polynomials. No matter how much they may insist.
Each term in a polynomial has what's called a degree, or a value based on the exponent attached to its variable. The degree of 9x2 is 2, for example. You may be unfamiliar with a degree of 2 unless you've ever been to Fairbanks, Alaska, in the middle of January.
We usually write the terms of a polynomial in descending order (greatest to least) according to the degree of each term. Exactly like a shuttle launch countdown. For example, we would write 2x2 + x instead of x + 2x2.
The degree of the constant 7 is zero, since 7 = 7x0. The degree of any other non-zero constant is also zero. However, the degree of the term 0 is undefined. You might want to get on that, Webster's.
Since 0 = 0x = 0x1 = 0x2 = 0x3 (all of these are the same as 0), any degree would work, and there's no obvious way to decide which one to pick. Mathematicians hate not knowing what to pick, so that's why they say it's "undefined." You should see them in a grocery store looking over grapefruit to see which is the most ripe. "Undefined...undefined...undefined...undefined..."
There's also a degree of each polynomial. You can figure out the degree of a polynomial if you haven't forgotten which numbers are bigger than each other. If all else fails, count them off on your fingers and hope you never run into anything bigger than 10.
Memorize this: the degree of a polynomial is the largest degree of any one term in the entire polynomial. While we usually write polynomials with the largest degree term first, it's a good idea to look at the degrees of all the terms, in case some impish degree sprite came along and mixed them up to make our lives miserable.
|5x3 + 6x + 9||3|
|x23 + x6 + 4x – 2||23|
|4x6 + 3x5 + 2x3||6|
|4x + 5x2||2|
The degree-0 term of a polynomial is also called the constant term of the polynomial—the number sitting all by itself, usually at the end of the polynomial. Who knows, maybe it couldn't find its deodorant this morning.
If a polynomial doesn't seem to have a constant term, as in 3x2 + 4x, we say its constant term is 0 because we can write "+ 0" at the end of any expression without changing the value of the expression. If you're ever asked to pay $10 + 0 for something, remember that it doesn't cost you anything extra before you decide to get all in a huff about it.
We can also have polynomials with more than one variable, called multivariable polynomials. They're like multivitamins, but even better for you. A multivariable polynomial is a finite sum of terms where each term looks like this:
(real #)(first variable)(first whole #)(second variable)(second whole #)…(last variable)(last whole #)
Not every variable needs to appear in every term. Some of them probably won't, since they were out partying late last night.
The following are multivariable polynomials.
The following all have multiple variables but are not multivariable polynomials, because they don't qualify as polynomials in the first place.
When finding the degree of a multivariable polynomial, remember to keep your head above ground. That goes for any ostriches who may be reading this. Ignore the constants and look for the exponents hovering in superscript. To find the degree of a multivariable term, add together the exponents of all the variables in that term. Isn't it nice to be asked to add once in a while?
You already know that the degree of a polynomial is the largest degree of any of its terms. Well, guess what? The same is true for multivariable polynomials. To see which term has the largest degree, we need to find the degree of each of the terms and then pick the biggest number. "Picking the biggest of something" is about the only thing easier than adding, so you should have no problems here.
We know what you're thinking. Aren't all polynomials special? Aw. That's sweet, but stop kissing up.
There are special names for polynomials with certain numbers of terms.
You can remember these three because a tricycle has three wheels, a bicycle has two wheels, and a monocycle has...man, that almost worked.
Another special kind of polynomial is a quadratic polynomial, which is a polynomial of degree 2. Yes, "quad" usually means "4," but bear with us.
A quadratic polynomial looks like ax2 + bx + c, where a, b, and c are real numbers and a isn't zero (if a were zero, the polynomial would only have degree 1).
A 2nd degree polynomial is "quadratic." Shouldn't it be "biratic?" What's the good of these numerical prefixes if they're only going to keep changing them on us?
A valid question. The answer? "Quad" also means "square." Oooh...sneaky.
In a single-variable polynomial of degree 2, we're squaring the variable, so it makes sense to think of that polynomial as "quadratic." We'll cross our fingers and hope "quad" doesn't also mean a third thing.
The following polynomials are quadratic.
The following polynomials are not quadratic.