Chapter 1: Algebraic Expressions and Polynomials

Fundamentals of Monomials and Polynomials

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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)
$\underbrace{-3}_{\color{#1976D2}{\text{coefficient}}} \;\underbrace{a^2b}_{\color{#F57C00}{\text{variable part}}}$
degree = 2 + 1 = 3
(exponent of $a$ + exponent of $b$)
Figure 1: Structure of the monomial $-3a^2b$. The coefficient is $-3$, the variable part is $a^2b$, and the degree is $2 + 1 = 3$.

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)
$3x^2$
quadratic term
$+$
$5x$
linear term
$-$
$2$
constant term
degree of polynomial = 2
(degree of the highest-degree term)
Figure 2: Structure of the polynomial $3x^2 + 5x - 2$. It has three terms; the degree of the highest-degree term $3x^2$ gives the polynomial degree $2$.

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.

  1. $-7x^3$
  2. $\dfrac{1}{2}a^2b^3$
  3. $5$

Problem 2

Simplify by combining like terms.

  1. $4x^2 - 3x + 2x^2 + 5x - 1$
  2. $2a + 3b - 5a + b$

Problem 3

Compute the following.

  1. $(2x^2 - 3x + 1) + (x^2 + 4x - 2)$
  2. $(5x^2 + 2x - 3) - (2x^2 - x + 1)$
Show Solutions

Problem 1 Solutions

  1. Coefficient: $-7$, degree: $3$
  2. Coefficient: $\dfrac{1}{2}$, degree: $5$ ($a$ twice + $b$ three times)
  3. Coefficient: $5$, degree: $0$

Problem 2 Solutions

  1. \begin{align} 4x^2 - 3x + 2x^2 + 5x - 1 &= (4x^2 + 2x^2) + (-3x + 5x) - 1 \\ &= 6x^2 + 2x - 1 \end{align}
  2. \begin{align} 2a + 3b - 5a + b &= (2a - 5a) + (3b + b) \\ &= -3a + 4b \end{align}

Problem 3 Solutions

  1. \begin{align} &(2x^2 - 3x + 1) + (x^2 + 4x - 2) \\ &= 2x^2 - 3x + 1 + x^2 + 4x - 2 \\ &= 3x^2 + x - 1 \end{align}
  2. \begin{align} &(5x^2 + 2x - 3) - (2x^2 - x + 1) \\ &= 5x^2 + 2x - 3 - 2x^2 + x - 1 \\ &= 3x^2 + 3x - 4 \end{align}