Polynomial Factorization
Introduction to Polynomials
What Are Polynomials?
In algebra, you'll often work with expressions made up of variables, coefficients, and exponents. When these components are combined using only addition, subtraction, and multiplication, you get a polynomial. The key rule is that the exponents of the variables must be non-negative whole numbers (like 0, 1, 2, 3, and so on). You can't have fractional or negative exponents.
A polynomial is an algebraic expression with one or more terms, where variables have only non-negative integer exponents.
Let's break down a typical polynomial to see its parts. Consider the expression $3x^2 - 4x + 7$.
Here's what each part means:
- Terms are the individual parts of the polynomial that are added or subtracted. In our example, the terms are , , and .
- Coefficients are the numerical parts of a term. For the term , the coefficient is 3. For , it's -4.
- A term that is just a number, like 7, is called a constant. You can think of it as , since any variable raised to the power of 0 is 1.
The Degree of a Polynomial
The degree of a term is the value of its exponent. In the term , the degree is 2. The term can be written as , so its degree is 1. A constant like 7 has a degree of 0.
The degree of the polynomial is the highest degree of any of its terms. For , the degrees of the terms are 2, 1, and 0. The highest one is 2, so the degree of the entire polynomial is 2.
For example, the polynomial $5x^4 + 2x^2 - 8$ has a degree of 4, because the highest exponent is 4.
The coefficient of the term with the highest degree is called the leading coefficient. In $5x^4 + 2x^2 - 8$, the leading coefficient is 5.
Naming Polynomials
Polynomials are often classified based on the number of terms they contain. This gives them special names that you'll see frequently.
| Name | Number of Terms | Example |
|---|---|---|
| Monomial | 1 | |
| Binomial | 2 | |
| Trinomial | 3 |
If a polynomial has more than three terms, we generally just call it a polynomial. For instance, $8x^4 - 2x^3 + x^2 - x + 1$ is a polynomial with five terms.
Polynomials can also be named by their degree. A polynomial of degree 1 is called linear, degree 2 is quadratic, and degree 3 is cubic. For now, focusing on the number of terms is most important.
Now, let's review these new terms.
Ready to check your understanding?
What is the leading coefficient of the polynomial ?
Which of the following expressions is NOT a polynomial?
Understanding these basic building blocks is the first step. Next, you'll learn how to perform operations with them.