Table of Contents
Understanding Polynomial Terms
In this chapter, we focus on the building blocks that make up polynomials: their individual terms. You will see what a polynomial term looks like, how to recognize and classify it, and how to work with the key parts of a term (coefficient, variable part, and exponent).
This chapter assumes you already know what variables, constants, and basic algebraic expressions are from earlier chapters.
What Is a Polynomial Term?
A polynomial term is an expression that is a product of:
- a number (the coefficient), and
- one or more variables raised to non-negative integer exponents.
Examples of single polynomial terms:
- $3x$
- $-5x^2$
- $\dfrac{1}{2}y^3$
- $7a^2b$
- $-4x^3y^2$
Non-examples (these are not polynomial terms):
- $x^{-1}$ (negative exponent)
- $3\sqrt{x}$, which is $3x^{1/2}$ (fractional exponent)
- $\dfrac{2}{x}$, which is $2x^{-1}$ (variable in the denominator)
The key restriction is: in a polynomial term, variable exponents must be whole numbers $0,1,2,3,\dots$.
Parts of a Polynomial Term
Consider the term
$$
-5x^3.
$$
It has three main parts:
- Coefficient: the numerical factor in front of the variables.
- In $-5x^3$, the coefficient is $-5$.
- In $7a^2b$, the coefficient is $7$.
- In $-\dfrac{3}{4}xy^2$, the coefficient is $-\dfrac{3}{4}$.
- Variable part: the product of variables with their exponents.
- In $-5x^3$, the variable part is $x^3$.
- In $7a^2b$, the variable part is $a^2b$.
- In $-\dfrac{3}{4}xy^2$, the variable part is $xy^2$.
- Sign: whether the term is positive or negative.
- The sign is often included as part of the coefficient.
- For example, $-5x^3$ has coefficient $-5$; $+2x$ has coefficient $2$.
A constant term is a term with no variables, such as $4$ or $-7$. You can think of its variable part as $x^0$ (since $x^0 = 1$), but we usually just write the number itself.
- In the polynomial $3x^2 - 2x + 5$, the constant term is $5$.
The Degree of a Term
The degree of a polynomial term tells you the “power” of the term. It is defined as:
- For a single-variable term (like $3x^4$): the degree is the exponent of the variable.
- Degree of $3x^4$ is $4$.
- Degree of $-7x$ (which is $-7x^1$) is $1$.
- Degree of a nonzero constant term (like $5$) is $0$.
- For a multi-variable term (like $4x^2y^3$): the degree is the sum of the exponents of all variables.
- Degree of $4x^2y^3$ is $2 + 3 = 5$.
- Degree of $-ab^2$ (that is $-a^1b^2$) is $1 + 2 = 3$.
- Degree of $6xyz^2$ is $1 + 1 + 2 = 4$.
If a term is $0$, we usually do not give it a degree; its degree is considered undefined in this context.
Monomials and Polynomial Terms
A monomial is simply one polynomial term considered as an expression by itself.
Examples of monomials:
- $2x^3$
- $-5$
- $7a^2b^3$
- $\dfrac{1}{3}y$
Every monomial is a polynomial term, and every polynomial term is a monomial.
A general polynomial is made by adding or subtracting these terms, but here we focus only on the individual terms, not on combining them into whole polynomials.
Like Terms
Polynomial terms are called like terms if they have the same variable part, meaning:
- the same variables, and
- each with the same exponents.
Only the coefficients may differ.
Examples:
- $3x^2$ and $-5x^2$ are like terms (same variable part $x^2$).
- $4ab$ and $-7ab$ are like terms.
- $\dfrac{1}{2}x^3y$ and $-6x^3y$ are like terms.
Non-like terms:
- $3x^2$ and $3x$ are not like terms (exponents differ: $2$ vs $1$).
- $4ab$ and $4a^2b$ are not like terms (exponent of $a$ differs).
- $5x^2y$ and $5xy^2$ are not like terms (exponents of $x$ and $y$ are swapped).
Recognizing like terms is important because only like terms can be combined (added or subtracted) into a single term.
Standard Form of a Term
Each polynomial term should be written in a standardized way to make it easy to compare and combine.
For a single term, standard form usually means:
- Write the coefficient first.
- Multiply by variables.
- Write variables in some fixed order (often alphabetical).
- Write exponents clearly and simplify products.
For example:
- $x \cdot 3x^2$ should be written as $3x^3$.
- $-2y \cdot y^3$ should be written as $-2y^4$.
- $5ab^2a$ should be written as $5a^2b^2$ (combine the $a$ factors and arrange as $a^2b^2$).
Be careful with signs:
- $-(3x^2)$ is $-3x^2$.
- $-1 \cdot -4x^3$ is $4x^3$.
Evaluating a Polynomial Term
To evaluate a polynomial term, substitute given values for the variables and perform the arithmetic.
Example: Evaluate $-3x^2y$ when $x = 2$ and $y = -1$.
- Substitute:
$$
-3x^2y \to -3(2)^2(-1).
$$ - Compute $2^2 = 4$:
$$
-3 \cdot 4 \cdot (-1).
$$ - Multiply:
$$
-12 \cdot (-1) = 12.
$$
So the value is $12$.
The process is the same for any polynomial term: plug in, follow exponent rules, then multiply.
Zero Coefficient and the Zero Term
If the coefficient of a term is $0$, the entire term is $0$, no matter what the variables are:
- $0x^5 = 0$
- $0xy^3 = 0$
The zero term (or just $0$) contains no variable part that matters. In polynomials, any term with a zero coefficient is effectively absent.
For example, $3x^2 + 0x + 5$ is just $3x^2 + 5$; the middle term is not really there.
Summary
- A polynomial term is a product of a numerical coefficient and variables raised to non-negative integer powers.
- Each term has a coefficient, a variable part, and a sign.
- The degree of a term is the exponent (single variable) or the sum of exponents (multiple variables).
- Like terms have exactly the same variable part; only coefficients may differ.
- Writing terms in standard form helps compare and combine them.
- Evaluating a term means substituting values for variables and carrying out the arithmetic.
- A zero coefficient makes the whole term equal to $0$, so that term does not affect the polynomial.