Polynomial Expressions 

A polynomial expression is a fundamental algebraic structure used throughout Precalculus and Calculus.
Polynomials are used to model motion, area, volume, profit, population trends, and many other real-world phenomena.

Understanding how polynomial expressions are built and manipulated is essential for:

• Polynomial functions and graphs

• Factoring and solving equations

• Analyzing end behavior

• Preparing for derivatives in Calculus

What Is a Polynomial Expression?

A polynomial expression is an algebraic expression consisting of:

• Constants

• Variables

• Non-negative integer exponents

• Addition, subtraction, and multiplication

General Form

$$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$​

Where:

• $$a_n,a_{n-1},\cdots\,a_0$$are real numbers (coefficients)

• n is a non-negative integer

• $$a_n\neq 0$$

What Is NOT a Polynomial?

Expressions are not polynomials if they contain:

• Variables in denominators

• Negative or fractional exponents

• Variables inside radicals

• Trigonometric, exponential, or logarithmic functions

Examples (Not Polynomials)

$$\frac{1}{x} , x^{-2} , \sqrt{x} , 3^{x}$$

Vocabulary of Polynomial Expressions

Term

A term is a product of a coefficient and variables with exponents.

Example:

$$5x^{3}$$

Coefficient

The coefficient is the numerical factor of a term.

$$-7x^2$$, the coefficient is −7.

Degree of a Term

The degree of a term is the sum of the exponents of the variables.

Example:

$$4x^3y^2 \longrightarrow$$ degree = 5

Degree of a Polynomial

The degree of a polynomial is the highest degree of its terms.

Example:

$$2x^4 – 5x^2 + 3 \longrightarrow$$ degree = 4

Classification of Polynomial Expressions

By Number of Terms

By Degree

Writing Polynomials in Standard Form

A polynomial is in standard form when:

• Terms are arranged in descending powers of the variable

• Like terms are combined

Example:

$$3x-2x^3+5+x^2 \rightarrow -2x^3+x^2+3x+5$$

Operations with Polynomial Expressions

Adding and Subtracting Polynomials

• Combine like terms only

Example:

$$(3x^2+2x-5) + (x^2-4x+1)=4x^2-2x-4$$

Multiplying Polynomials

Monomial × Polynomial

$$2x(3x^2-x+4)=6x^3-2x^2+8x$$

Binomial × Binomial

$$(x+3)(x-5)=x^2-2x-15$$

Factoring Polynomial Expressions

Factoring is the reverse of multiplication.

Common Factors

$$6x^3+9x^2=3x^2(2x+3)$$

Factoring Trinomials

$$x^2+5x+6=(x+2)(x+3)$$

Difference of Squares

$$x^2-9=(x-3)(x+3)$$

Polynomial Expressions and Functions

A polynomial expression becomes a polynomial function when written as:

f(x)=polynomial expression

This allows:

• Graphing

• Finding zeros

• Analyzing end behavior

• Studying rates of change

The Binomial Theorem

The Binomial Theorem provides a systematic method for expanding powers of a binomial expression of the form

$$(a+b)^n$$

without multiplying the expression repeatedly.

This theorem is essential in Precalculus because it:

• Simplifies expansion of high-degree polynomials

• Reveals patterns in coefficients

• Connects algebra with combinatorics and probability

• Prepares students for series and approximations in Calculus

Binomial Expressions

A binomial is a polynomial with exactly two terms.

Examples:

$$x+3 , 2a-b , 1-4x$$

Powers of binomials appear frequently in algebraic modeling and function analysis.

Pattern in Small Powers

Consider the expansions:

$$(a+b)^1=a+b$$

$$(a+b)^2=a^2+2ab+b^2$$

$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$

$$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$

Observations:

• Powers of $a$ decrease

• Powers of $b$ increase

• Coefficients follow a predictable pattern

Statement of the Binomial Theorem

For any positive integer n:

$$(a+b)^n=\sum\limits_{k=0}^n\left(_k^n\right)a^{n-k}b^k$$

Where:

• $$\left(_k^n\right)$$ is a binomial coefficient

• k represents the position of each term

Binomial Coefficients

Definition

$$\left(_k^n\right)=\frac{n!}{k!(n-k)!}$$​

These coefficients count the number of ways to choose k objects from n.

Example

$$\left(_2^5\right)=\frac{5!}{2!(5-2)!}=10$$

Pascal’s Triangle

Binomial coefficients appear in Pascal’s Triangle.

1

1   1

1   2   1

1   3   3   1

1   4   6   4   1

$$\cdots$$

Each row corresponds to the coefficients of $$(a+b)^n$$.

Example: Full Expansion

Expand:

$$(x+2)^4$$

Using binomial coefficients 1,4,6,4,1:

$$(x+2)^4=x^4+4x^3(2)+6x^2(2)^2+3x(2)^3+2^4$$

$$=x^4+8x^3+24x^2+32x+16$$

Finding a Specific Term

Find the 3rd term of:

$$(2x-3)^5$$

Use:

$$T_3=\left(_2^5\right)(2x^3)(-3)^2=10\cdot8x^3\cdot9=720x^3$$