Polynomials

Table of Contents

Understanding Polynomials

A polynomial is an algebraic expression built using only:

variables
numbers (constants)
the operations of addition, subtraction, and multiplication
and non-negative integer exponents on the variables

So a typical polynomial in $x$ might look like
$$
4x^3 - 2x^2 + 7x - 5.
$$

Expressions that use division by a variable (like $\frac{3}{x}$ or $x^{-1}$) or roots of a variable (like $\sqrt{x}$) are not polynomials.

Terms, Coefficients, and Degrees (Overview)

Polynomials are made by putting together terms. A term is a product of:

a number (the coefficient), and
one or more variables raised to non-negative integer powers.

For example, in
$$
-6x^4y^2 + 3x^2 - 8,
$$

$-6x^4y^2$ is a term,
$3x^2$ is a term,
$-8$ is a term (a constant term).

The coefficient is the numerical part of the term:

$-6$ is the coefficient of $x^4y^2$,
$3$ is the coefficient of $x^2$,
$-8$ is the constant term (you can think of it as $-8x^0$).

The degree of a term:

In one variable $x$, the degree is the exponent of $x$.
Example: x^5$ has degree $.
In more than one variable, the degree of a term is the sum of the exponents.
Example: x^3y^2$ has degree +2 = 5$.

The degree of a polynomial is the highest degree of any of its terms (once it’s simplified and like terms are combined).

Examples:

$5x^3 - 4x + 1$

Term degrees: $3, 1, 0$
Polynomial degree: $3$

$7x^2y^3 + x^4 - 9$

Term degrees: $2+3 = 5$, $4$, $0$
Polynomial degree: $5$ (from $7x^2y^3$)

Standard Form and Naming by Degree

A polynomial in one variable is usually written in standard form, which means:

Write the terms in order of descending degree, from highest power to lowest.
Combine any like terms (terms with the same variable part).

Example:
$3 - x^2 + 5x + 4x^2$

Combine like terms: $-x^2 + 4x^2 = 3x^2$
Then write in descending powers:
$$
3x^2 + 5x + 3
$$

Polynomials in one variable have specific names based on their degree:

Degree $0$: constant polynomial (e.g. $7$)
Degree $1$: linear polynomial (e.g. $3x - 2$)
Degree $2$: quadratic polynomial (e.g. $x^2 - 5x + 6$)
Degree $3$: cubic polynomial (e.g. $2x^3 + x - 1$)
Degree $4$: quartic polynomial
Degree $5$: quintic polynomial

We can also name a polynomial by its number of terms:

1 term: monomial (e.g. $4x^3$)
2 terms: binomial (e.g. $x - 5$)
3 terms: trinomial (e.g. $x^2 + 3x + 2$)

So $2x^3 - 7$ is a cubic binomial, and $x^2 + 3x + 1$ is a quadratic trinomial.

What Is and Isn’t a Polynomial

To decide whether an expression is a polynomial, check two main things:

The variable exponents must be whole numbers (0, 1, 2, 3, …).
There must be no division by a variable.

Examples of polynomials:

$5x^4 - 3x + 1$
$-2x^2y + 7y^3$
$9$ (a constant polynomial)
$\frac{1}{2}x^7 - \sqrt{3}x$ (coefficients can be any real numbers)

Not polynomials:

$\displaystyle \frac{3}{x}$ (this is $3x^{-1}$, exponent is negative)
$\sqrt{x} + 1$ (this is $x^{1/2} + 1$, exponent is not an integer)
$\displaystyle \frac{2}{x^2 + 1}$ (variable is in the denominator)
$\displaystyle x^{-3} + 4$ (negative exponent)

Polynomials can have fractions or square roots as coefficients, just not as exponents on the variables. For instance,
$$
\left(\frac{1}{\sqrt{2}}\right)x^3 - \frac{5}{7}x
$$
is still a polynomial in $x$.

Evaluating Polynomials

To evaluate a polynomial, substitute a given number for the variable and simplify.

Example:
Evaluate $P(x) = 2x^3 - 3x + 4$ at $x = -2$.

Substitute $x = -2$:
$$
P(-2) = 2(-2)^3 - 3(-2) + 4
$$
Compute step by step:

$(-2)^3 = -8$ so $2(-8) = -16$
$-3(-2) = 6$
So
$$
P(-2) = -16 + 6 + 4 = -6
$$

The idea of evaluating will appear again when working with roots, graphs, and functions, but here you just need to be comfortable plugging in and simplifying carefully (especially with signs and parentheses).

Polynomials in Several Variables (Brief View)

A polynomial can involve more than one variable, such as $x$ and $y$:
$$
Q(x,y) = 3x^2y - 4y^3 + 2.
$$

Key points:

Each term is still a product of a coefficient and variables with non-negative integer exponents.
The degree of a term is the sum of exponents in that term.
The degree of the polynomial is the largest degree among its terms.

For $Q(x,y)$ above:

$3x^2y$ has degree $2+1 = 3$
$-4y^3$ has degree $3$
$2$ has degree $0$
So $Q$ has degree $.

You handle addition and multiplication of such polynomials using the same basic ideas as for one variable; the presence of more variables just means you have more kinds of like terms to watch for.

Why Polynomials Matter in Algebra I

Polynomials are central objects in algebra because:

Many functions you study are polynomials.
Equations like $x^2 - 5x + 6 = 0$ or $2x^3 + x - 1 = 0$ are polynomial equations.
Later chapters on polynomial operations, factoring, and quadratic equations all build on the basic understanding of what a polynomial is and how it is structured.

For now, in this chapter, the essential skills are:

Recognizing polynomial expressions.
Distinguishing polynomials from non-polynomials.
Identifying terms, coefficients, and degrees.
Writing polynomials in standard form.
Evaluating a polynomial at a given value.