Chapter 1: Algebraic Expressions and Polynomials
Algebra Introduction - Chapter 1 / Fundamentals of Monomials and Polynomials
Introductory (high school level)
What Are Algebraic Expressions?
In mathematics, we often use variables (letters) in place of specific numbers to represent quantities and expressions. This allows us to state general rules concisely.
The Advantage of Using Variables
If we let the height be $a$, the width be $b$, and the area be $S$, then "area of a rectangle = height × width" can be written simply as
$$S = ab$$Moreover, by substituting any lengths for $a$ and $b$, we can compute the area of any rectangle.
Monomials
Monomial
An expression consisting only of a product of numbers and variables is called a monomial.
Examples: $5x$, $-3a^2b$, $\dfrac{2}{3}xy^2$, $7$, $a$ (a number alone or a single variable is also a monomial)
Coefficient and Degree
- Coefficient: the numerical part of a monomial
- Degree: the total number of variable factors. A nonzero constant (e.g. $7$) has degree $0$
Note: The degree of $0$ is left undefined. This is because $0$ can be written as a product with any degree ($0 = 0 \cdot x = 0 \cdot x^2 = \cdots$), so assigning a specific degree would lead to contradictions.
Examples: Coefficient and Degree of Monomials
| Monomial | Coefficient | Degree |
|---|---|---|
| $5x$ | $5$ | $1$ |
| $-3a^2b$ | $-3$ | $3$ ($a$ appears twice, $b$ once) |
| $\dfrac{2}{3}xy^2$ | $\dfrac{2}{3}$ | $3$ ($x$ once, $y$ twice) |
| $7$ | $7$ | $0$ (constant term) |
Polynomials
Polynomial
An expression written as a sum of monomials is called a polynomial.
Each monomial in a polynomial is called a term.
Example: A Polynomial
$$3x^2 + 5x - 2$$This polynomial consists of three terms:
- $3x^2$ (quadratic term, degree 2)
- $5x$ (linear term, degree 1)
- $-2$ (constant term, degree 0)
Degree of a Polynomial
The degree of a polynomial is the highest degree among all its terms.
Examples: Degree of Polynomials
- $3x^2 + 5x - 2$ has degree $2$ (the highest-degree term is $3x^2$)
- $x^3 - 4x + 1$ has degree $3$
- $5$ has degree $0$ (constant only)
Like Terms
Like Terms
Terms whose variable parts are exactly the same are called like terms.
Like terms can be combined by adding or subtracting their coefficients.
Example: Combining Like Terms
\begin{align} 3x^2 + 5x - 2x^2 + 4x &= (3x^2 - 2x^2) + (5x + 4x) \\ &= (3 - 2)x^2 + (5 + 4)x \\ &= x^2 + 9x \end{align}Identifying Like Terms
- $5x$ and $4x$ → like terms (variable part $x$ is the same)
- $3x^2$ and $-2x^2$ → like terms (variable part $x^2$ is the same)
- $2xy$ and $3yx$ → like terms ($xy = yx$)
- $3x^2$ and $5x$ → not like terms (degrees differ)
- $3x$ and $5y$ → not like terms (variable parts differ)
Ordering Polynomials
Polynomials are usually arranged in descending order of degree (from the highest-degree term to the lowest).
Example: Ordering a Polynomial
Arrange $5 - 3x + 2x^3 + x^2$ in descending powers of $x$:
$$2x^3 + x^2 - 3x + 5$$Example: A Polynomial in Two Variables
Arrange $3xy + 2x^2 - y^2 + 5x - 1$ in descending powers of $x$:
\begin{align} &= 2x^2 + (3y + 5)x + (-y^2 - 1) \\ &= 2x^2 + (3y + 5)x - y^2 - 1 \end{align}Viewed as a quadratic in $x$, the coefficients involve $y$.
Addition and Subtraction of Polynomials
Addition of Polynomials
To add polynomials, combine the coefficients of like terms.
Example: Addition
Compute $(3x^2 + 2x - 1) + (x^2 - 5x + 4)$:
\begin{align} &(3x^2 + 2x - 1) + (x^2 - 5x + 4) \\ &= 3x^2 + 2x - 1 + x^2 - 5x + 4 \\ &= (3x^2 + x^2) + (2x - 5x) + (-1 + 4) \\ &= 4x^2 - 3x + 3 \end{align}Subtraction of Polynomials
To subtract a polynomial, change the sign of every term in the polynomial being subtracted, then add.
Example: Subtraction
Compute $(3x^2 + 2x - 1) - (x^2 - 5x + 4)$:
\begin{align} &(3x^2 + 2x - 1) - (x^2 - 5x + 4) \\ &= 3x^2 + 2x - 1 - x^2 + 5x - 4 \\ &= (3x^2 - x^2) + (2x + 5x) + (-1 - 4) \\ &= 2x^2 + 7x - 5 \end{align}Caution: Removing Parentheses
When expanding $-(x^2 - 5x + 4)$, every term changes sign:
$$-(x^2 - 5x + 4) = -x^2 + 5x - 4$$Note that $-5x$ becomes $+5x$, and $+4$ becomes $-4$.
Exercises
Problem 1
Find the coefficient and degree of each monomial.
- $-7x^3$
- $\dfrac{1}{2}a^2b^3$
- $5$
Problem 2
Simplify by combining like terms.
- $4x^2 - 3x + 2x^2 + 5x - 1$
- $2a + 3b - 5a + b$
Problem 3
Compute the following.
- $(2x^2 - 3x + 1) + (x^2 + 4x - 2)$
- $(5x^2 + 2x - 3) - (2x^2 - x + 1)$
Show Solutions
Problem 1 Solutions
- Coefficient: $-7$, degree: $3$
- Coefficient: $\dfrac{1}{2}$, degree: $5$ ($a$ twice + $b$ three times)
- Coefficient: $5$, degree: $0$
Problem 2 Solutions
- \begin{align} 4x^2 - 3x + 2x^2 + 5x - 1 &= (4x^2 + 2x^2) + (-3x + 5x) - 1 \\ &= 6x^2 + 2x - 1 \end{align}
- \begin{align} 2a + 3b - 5a + b &= (2a - 5a) + (3b + b) \\ &= -3a + 4b \end{align}
Problem 3 Solutions
- \begin{align} &(2x^2 - 3x + 1) + (x^2 + 4x - 2) \\ &= 2x^2 - 3x + 1 + x^2 + 4x - 2 \\ &= 3x^2 + x - 1 \end{align}
- \begin{align} &(5x^2 + 2x - 3) - (2x^2 - x + 1) \\ &= 5x^2 + 2x - 3 - 2x^2 + x - 1 \\ &= 3x^2 + 3x - 4 \end{align}
Frequently Asked Questions
Q1: What is the difference between a monomial and a polynomial?
A: A monomial is an expression written only as a product of numbers and variables (e.g. $5x$, $-3a^2b$), whereas a polynomial is written as a sum of monomials (e.g. $3x^2 + 5x - 2$). In English usage a monomial simply counts as a polynomial with a single term.
Q2: What are the coefficient and degree of a monomial?
A: The coefficient is the numerical part of a monomial, and the degree is the number of variable factors multiplied together. For example, $-3a^2b$ has coefficient $-3$ and degree $2+1=3$ (two $a$'s and one $b$). The degree of $0$ is left undefined.
Q3: What are like terms?
A: Terms whose variable parts are exactly the same are called like terms. For example, $3x^2$ and $-2x^2$ are like terms, so only their coefficients are added or subtracted, giving $x^2$. Thus $3x^2 + 2x - x^2 + 5x = 2x^2 + 7x$. By contrast, $3x^2$ and $5x$ are not like terms because their variable parts differ.