What is Polynomial?
An algebraic expression that contains variable terms and constant terms. The terms are separated by addition and subtraction signs, and the exponents of the variables are non-negative integers.
A polynomial is a type of algebraic expression made up of terms that are added or subtracted. These terms can include:
- Constants (like 5 or –2)
- Variables (like x or y)
- Coefficients (the numbers in front of the variables)
Polynomials can look like this:
- 3x – 1
- 3y²+5
- 2x + 3xy
- 3xy – 2
The key rule for polynomials is that the exponents of the variables must be non-negative integers (0, 1, 2, 3, ...). This means you won’t see variables with negative exponents, fractional exponents, or variables in the denominator.
So, polynomials cannot look like this:
- \(\displaystyle 5x^{-2} + 3\) because the exponent –2 is negative, which is not allowed.
- \(\displaystyle \frac{4x + 2}{x}\) because the variable x is in the denominator, which is not allowed.
- \(\displaystyle 7\sqrt{x} + 2\) because the square root of x is equivalent to \(\displaystyle x^{\frac{1}{2}}\), which is a fractional exponent and not allowed for polynomials.
Polynomials can have different numbers of terms, which is a way to classify them into:
- Monomials: Polynomials with one term (e.g., 5x or 3y²)
- Binomials: Polynomials with two terms (e.g., x + 2 or 3x – 5)
- Trinomials: Polynomials with three terms (e.g., x² + 3x + 4)
And what do we call polynomials with more than three terms? Simply, polynomials.
Polynomials are powerful because they can represent many different types of problems, including:
- Area and volume calculations
- Predicting trends in data
- Solving real-world problems in science and engineering
When Do Students Learn About Polynomials?
Students are introduced to polynomials when they begin learning about algebraic expressions. This typically starts in middle school and continues through high school.
Grades 6–8 – Introduction to Polynomials
Students learn to identify, add, and subtract polynomials. They also begin practicing simple multiplication of polynomials.
Grades 9+ – Working with Polynomials in Algebra
Students learn to multiply, divide, and factor polynomials. They explore how polynomials are used in graphing, solving equations, and understanding real-world applications.
