What Is a Polynomial

Write down the expression $4x^3 - x + 9$. Now write $4x^3 - x^{-1} + 9$. The two differ by a single minus sign in an exponent. One of them can be evaluated at every real number and the other cannot. Which is which, and what goes wrong at $x = 0$?

A polynomial in $x$ is a sum of terms of the form a real number times a whole-number power of $x$. That is the whole definition. Everything else in this focus area is vocabulary for describing one, or a technique for taking one apart.

A polynomial in $x$ is any expression that can be written as

$$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$$

where $n$ is a whole number and $a_0, a_1, \ldots, a_n$ are real numbers. Two conditions are doing all the work: the powers of $x$ are whole numbers, and the coefficients are ordinary real numbers.

So these are polynomials:

$$x^2 + 3x, \qquad 7 - 2x^5, \qquad 5, \qquad 0, \qquad \tfrac{2}{3}x^3 - \sqrt{2}\,x$$

The last two are worth a look. A plain number like $5$ is a polynomial, because $5 = 5x^0$ and $0$ is a whole number. And $\sqrt{2}$ is fine as a coefficient: the rule restricts the powers of $x$, not the numbers in front.

These are not polynomials:

$$\frac{1}{x} = x^{-1}, \qquad \sqrt{x} = x^{1/2}, \qquad 2^x, \qquad |x|, \qquad \frac{x+1}{x-3}$$

Each fails the same test once rewritten. A negative power, a fractional power, the variable in the exponent, a shape that no single power produces, and a variable in a denominator.

The degree of a polynomial is the highest power of $x$ appearing with a non-zero coefficient. Write the polynomial in descending powers and it is the first exponent you see.

$$P(x) = 3x^4 - 5x^2 + 7x - 2 \quad \text{has degree } 4$$

Degree is not just a label. It controls almost everything you will meet later: how many zeroes the polynomial can have, how many times its graph can turn, what happens far out to the left and right, and how long a division will take. Low degrees have names you already use:

degree $0$ is a non-zero constant, degree $1$ is linear, degree $2$ is quadratic, degree $3$ is cubic, degree $4$ is quartic.

Two edge cases are worth settling now. A non-zero constant such as $5$ has degree $0$, since $5 = 5x^0$. The zero polynomial $P(x) = 0$ is given no degree at all, because there is no non-zero coefficient to point at. That sounds like pedantry until Lesson 3, where the degree of a product is the sum of the degrees, and the zero polynomial would break the rule.

Take $P(x) = 3x^4 - 5x^2 + 7x - 2$ and name every part.

The leading term is $3x^4$, the term of highest degree. Its coefficient, $3$, is the leading coefficient. The constant term is $-2$, the term with no $x$.

The coefficient of $x^2$ is $-5$, not $5$. A coefficient carries the sign of its term, and losing that sign is the single most common slip in this topic.

Two more traps. The coefficient of $x^3$ here is $0$, because no $x^3$ term is written; a missing term is a zero coefficient, not an absent one, which will matter enormously when you set up a division in Lesson 5. And the coefficient of $x$ is $7$, since $7x$ means $7x^1$.

Writing out every power, including the ones with coefficient zero, gives

$$P(x) = 3x^4 + 0x^3 - 5x^2 + 7x - 2$$

which is the same polynomial, written in the form that makes every coefficient visible.

A polynomial is monic when its leading coefficient is exactly $1$.

$$x^3 - 4x + 1 \quad \text{is monic} \qquad 2x^3 - 4x + 1 \quad \text{is not}$$

Three things people get wrong about this word:

It is about the leading coefficient only. $x^3 + 6x^2 - 11$ is monic even though $6$ and $-11$ are nowhere near $1$.

A leading coefficient of $-1$ makes it non-monic. $-x^2 + 3$ has leading coefficient $-1$, and $-1 \neq 1$.

Monic is not a property of the polynomial's value. $2x + 2$ is non-monic, even though it factors as $2(x+1)$ where $x + 1$ is monic. Dividing out the leading coefficient always produces a monic polynomial, and that is a genuinely useful move, but it changes the polynomial rather than revealing that it was monic all along.

Five quick items on the definition and the vocabulary. Work each one, then reveal the answer.

1. 1 Is $5x^2 - \dfrac{3}{x^2}$ a polynomial?

   Show answer
   The second term is $-3x^{-2}$, and $-2$ is not a whole number.No

2. 2 State the degree and leading coefficient of $8 - 3x^4 + x$.

   Show answer
   Reorder: $-3x^4 + x + 8$.Degree $4$, leading coefficient $-3$

3. 3 What is the coefficient of $x^3$ in $x^5 - 2x^2 + 6$?

   Show answer
   No $x^3$ term is written, so its coefficient is zero.$0$

4. 4 Is $-x^7 + x$ monic?

   Show answer
   Its leading coefficient is $-1$, and monic requires exactly $+1$.No

5. 5 What is the degree of the constant polynomial $P(x) = 12$?

   Show answer
   $12 = 12x^0$, so the highest power present is zero.$0$