BackPolynomials: Vocabulary, Addition, and Subtraction
Study Guide - Smart Notes
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Understanding the Vocabulary Used to Describe Polynomials
Definition of a Polynomial
A polynomial is a single term or the sum of two or more terms containing variables with whole-number exponents. Polynomials are fundamental objects in algebra and are used to model a wide variety of real-world and mathematical situations.
Term: A part of a polynomial separated by plus or minus signs. Each term consists of a coefficient and a variable raised to a power.
Variable: A symbol, usually x, representing an unknown value.
Exponent: The power to which the variable is raised. In polynomials, exponents are always whole numbers.
Example polynomial:
This polynomial can be expressed as:
The polynomial contains four terms.
It is customary to write the terms in the order of descending powers of the variables.
The degree of a polynomial is the highest exponent of the variable in the polynomial.
Degree of a Polynomial
The degree of a nonzero constant is 0.
The constant 0 has no defined degree.
Great Question: If the degree of a nonzero constant is 0, why doesn't the constant 0 also have degree 0?
We can express 0 in many ways, including , , and . It is impossible to assign a unique exponent to the variable. This is why 0 has no defined degree.
The Degree of : If and is a whole number, the degree of is . The degree of a nonzero constant term is 0. The constant 0 has no defined degree.
Types of Polynomials by Number of Terms
Monomial: A polynomial with one term (e.g., ).
Binomial: A polynomial with two terms (e.g., ).
Trinomial: A polynomial with three terms (e.g., ).
Polynomials with four or more terms have no special names.
The degree of a polynomial is the greatest degree of all its terms. For example, is a binomial of degree 2.
Polynomial | Number of Terms | Degree | Special Name |
|---|---|---|---|
3 | 2 | Trinomial | |
4 | 4 | None | |
2 | 2 | Binomial |
Adding Polynomials
Combining Like Terms
To add polynomials, combine like terms—terms containing exactly the same variables raised to the same powers. This is done by adding the coefficients of the like terms.
Example:
Add
Combine like terms for , , and constants.
Subtracting Polynomials
Finding the Opposite of a Polynomial
The opposite of a polynomial is that polynomial with the sign of every coefficient changed. To subtract one polynomial from another, add the opposite of the polynomial being subtracted.
Rule: To subtract two polynomials, add the first polynomial and the opposite of the second polynomial.
Example:
Subtract
Change the sign of each term in the second polynomial, then add to the first.
Examples of Subtracting Polynomials
Subtract from .
Subtract from .
Vertical Subtraction of Polynomials
Polynomials can also be subtracted by aligning like terms in columns and subtracting vertically, similar to how numbers are subtracted.
Example:
Subtract using the column method.
Summary Table: Key Properties of Polynomials
Property | Description |
|---|---|
Term | A part of a polynomial separated by + or - signs |
Degree | Highest exponent of the variable in the polynomial |
Like Terms | Terms with the same variable(s) and exponent(s) |
Monomial | Polynomial with one term |
Binomial | Polynomial with two terms |
Trinomial | Polynomial with three terms |
Additional info:
When adding or subtracting polynomials, always arrange terms in descending order of degree for clarity.
Polynomials are foundational for later topics such as factoring, solving equations, and graphing.
