Algebra: Monomials and Polynomials
What is a Monomial?
A monomial is a single term algebraic expression that consists of a constant, a variable, or the product of a constant and one or more variables. A monomial does not include addition, subtraction, or division by variables within the term. The variables in a monomial are raised to non-negative integer exponents.
Examples of Monomials:
- Constant Only: \(7\)
A simple constant with no variable is a monomial.
- Single Variable: \(x\)
A single variable by itself is a monomial.
- Constant and Variable: \(4y\)
The product of a constant and a variable.
- Multiple Variables: \(3ab\)
The product of a constant and two variables.
- Variable with Exponent: \(2x^3\)
A variable raised to a positive integer exponent.
- Multiple Variables with Exponents: \(5x^2y^3\)
The product of a constant and variables, each raised to a positive integer exponent.
Examples that are NOT Monomials:
- Addition or Subtraction : \(x + 2y\)
This is a binomial because it has two terms added together.
- Division by a Variable: \(\frac{3}{x}\)
Dividing by a variable disqualifies it from being a monomial.
- Negative Exponent:\(4x^{-2}\)
Monomials do not include negative exponents.
- Root of a Variable:\(\sqrt{x}\)
Taking the square root of a variable is not allowed in a monomial as it implies a fractional exponent.
Key Points:
- A monomial can be as simple as a number or a single variable, or as complex as the product of several variables and constants.
- • The defining characteristic of a monomial is that it is a single term with variables having non-negative integer exponents.
What is a Polynomial?
A polynomial is an algebraic expression composed of one or more terms, where each term is a product of a constant (called the coefficient) and one or more variables raised to non-negative integer exponents. The terms are connected by addition or subtraction. The degree of a polynomial is the highest exponent of the variable in the expression.
Examples of Polynomials:
| Polynomials |
Coefficients (separated by comma) |
Degree |
| \(7\) |
7 |
0 |
| \(x + 3\) |
1, 3 |
1 |
| \(2x^2 + 5x + 1\) |
2, 5, 1 |
2 |
| \(-x^3 + 4x^2 - 7x + 2\) |
-1, 4, -7, 2 |
3 |
| \(6x^4 - 2x^3 + x - 9\) |
6, -2, 1, -9 |
4 |
Key Points:
- Coefficients: These are the numerical factors of the terms in the polynomial.
- Degree: The highest power (exponent) of the variable in the polynomial.
- Polynomial Types: Polynomials can be classified by the number of terms (monomial, binomial, trinomial) and by the degree of the polynomial.
Examples that are NOT Polynomials:
- Negative Exponents:
Expression: \(x^{-2} + 3x + 1\)
Reason: Polynomials cannot have variables with negative exponents.
- Fractional Exponents:
\(4x^{1/2} + 2x\)
Reason: Polynomials cannot have variables raised to fractional exponents (e.g., \(x^{1/2}\) is the square root of x).
- Variables in the Denominator:
\(\frac{3}{x} + 2x\)
Reason: Polynomials cannot have variables in the denominator of any term.
- Roots of Variables:
\(\sqrt{x} + x^2 + 5\)
The square root of x is equivalent to \(x^{1/2}\), which is not allowed in polynomials.
- Trigonometric Functions:
\(2x + \sin(x)\)
Trigonometric functions like sin(x), exponential functions like \(e^{x}\), or logarithmic functions like log(x) are not part of polynomial expressions.