Polynomials | Putting the "Why?" in Math

Advanced Functions is HARD . Everyone says it, so it must be true… right?

But why is it hard? Sure, the symbols get confusing, but the quadratic formula looks like witchcraft to the average 8th grader, yet we know it’s just a tool. In fact, the quadratic formula is a shortcut ; it just looks complicated at first glance.

I think that basic idea represents all of math: the more you learn, the more shortcuts for older material you will find.

Math isn’t hard, it just looks hard.

Advanced Functions is all of that, just more. It’s not a new subject—it’s the DLC to Functions .

To get started on this DLC, let’s head back to a topic we all know and “love”: the Polynomials .

What is a Polynomial?

A Polynomial is simply an expression with one or more terms added or subtracted.

The name is a clue! It literally means “many terms.” This is why we have special names for the ones with fewer terms:

Monomial: One term ( mono- )

Binomial: Two terms ( bi- )

Trinomial: Three terms ( tri- )

Anything with one or more terms is technically a polynomial. They are the simple, reliable building blocks we need for this journey.

The Anatomy of a Term

Okay, so a polynomial is made of terms. But what is a term, really? A single term, like 5x³, has a few key parts that we’ll see over and over.

The Coefficient : This is the number in front, like the 5. It’s the multiplier.
The Variable : This is the letter part that can change, like the x.
The Exponent : This is the small number up top, like the ³, indicating the power.

Together, they form a single package deal: 5 x ³ . When you string these together with addition or subtraction, you get a full-blown polynomial, like 5x³ - 2x + 7.

The One Big Rule

There’s one crucial rule that separates a true polynomial from a look-alike, and it’s all about the exponents.

The exponents on the variables must be whole numbers (0, 1, 2, 3, …).

No fractions. No negatives. If the exponent isn’t a nice, clean, non-negative integer, it’s not a polynomial.

These ARE Polynomials:

7x² + 3 (Exponents are 2 and 0, both are whole numbers.)
x⁵ - 2x⁴ + x (Exponents are 5, 4, and 1.)
12 (This is really 12x⁰. The exponent is 0, which is a whole number!)

These are NOT Polynomials:

3x⁻² + 1 (The -2 is negative, which is not allowed.)
√x + 5 (A square root is a fractional exponent: x¹/². Not a whole number.)
4/x - x² (A variable in the denominator is a negative exponent: 4x⁻¹. No go.)

So, if you see negative exponents, fractional exponents, or variables in the denominator, you’re not looking at a polynomial. It’s a different beast, like a Rational Function , which we’ll tackle later in the DLC.