- Topics At a Glance
- Exponents
- Negative Exponents
- Fractional Exponents
- Irrational Exponents
- Variables as Exponents
**Defining Polynomials**- Degrees of a Polynomial
- Multivariable Polynomials
- Degrees of Multivariable Polynomials
- Special Kinds of Polynomials
- Evaluating Polynomials
- Roots of a Polynomial
- Combining Polynomials
- Multiplying Polynomials
- Multiplication of a Monomial and a Polynomial
- Multiplication of Two Binomials
- Special Cases of Binomial Multiplication
- General Multiplication of Polynomials
- We'll Divide Polynomials Later!
- Factoring Polynomials
- The Greatest Common Factor
- Recognizing Products
- Trial and Error
- Factoring by Grouping
- Summary
- Introduction to Polynomial Equations
- Solving Polynomial Equations
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.

We determine what kind of polynomial we have by looking at its parts, or **terms**. Terms like 2*x* 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). 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.

is a polynomial.

Where is 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), the last term should be ** x ^{0 }= 1 and...thought of as 4x^{0}**

which is a real number multiplied by a whole number power of *x*.

4*x*^{2} – 3*x* – 1 is also polynomial.

The first term of this polynomial is 4*x*^{2}. Since we could rewrite the polynomial as

4*x*^{2} + (-3)*x* + (-1),

the second term of this polynomial is -3*x* and the third term is -1. Watch those negative signs. They'll getcha.

In fact, the following are all polynomials.

*x*+ 4

*x*^{2}+ 2*x*– 2

- 2
*x*^{34}– 4.5*x*^{23}+*x*^{3}+ 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 to be authentic polynomials. 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

- 4
*x*^{(¾)}is not a polynomial because*x*is being raised to a power that is not a whole number.

- 5
*x*^{7}+ 2is not a polynomial because^{x}*x*is not allowed to occur as an exponent in a polynomial. Get down please,*x*. Don't make us come up there.

Exercise 1

Is the following expression a polynomial or not? If not, explain why:

5*x*^{2} + 3*x*^{-4}

Exercise 2

Is the following expression a polynomial or not? If not, explain why:

4

Exercise 3

Is the following expression a polynomial or not? If not, explain why:

Exercise 4

Is the following expression a polynomial or not? If not, explain why:

Exercise 5

Is the following expression a polynomial or not? If not, explain why:

Exercise 6

List the terms in the following polynomial:

*x* + 7

Exercise 7

List the terms in the following polynomial:

-2*x*^{3} - 3*x* + 1

Exercise 8

List the terms in the following polynomial:

4*x*^{2} + 2*x* - 1