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.
- x + 4
- x2 + 2x – 2
- 2x34– 4.5x23 + x3 + 5.3
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.
is not a polynomial because 
. In a polynomial, we're not allowed to raise x to a negative exponent. We tried once and we totally got our knuckles thwacked with a ruler. #darkages
- 4x(¾) is not a polynomial because x is being raised to a power that's not a whole number.
- 5x7 + 2x is not a polynomial because x is not allowed to occur as an exponent in a polynomial. Get down please, x. Don't make us come up there.