There are several types of algebraic expressions in algebra. These expressions are characterized by their number of terms.
Monomials are algebraic expressions containing a single term. The terms can be constant terms or variables.
|4,6xy^2z^3,| and |34d| are all monomials.
Polynomials are algebraic expressions containing one or more terms. In fact, a polynomial is the algebraic sum or difference of multiple monomials.
The word polynomial is commonly used to designate expressions containing several terms. These terms can be constant or algebraic.
|2ab-3r+9u+xy-7| is a polynomial.
|x^3+6s^2t-4x+5t+2| is a polynomial.
Polynomials can contain one or more variables. A single-variable polynomial is a combination of several terms containing only one unique variable, or letter. In contrast, a multivariable polynomial is a collection of terms containing several variables.
Consider the following expression. ||\color{green}{x}^3+\color{green}{x}^2-3\color{green}{x}+6|| It is a polynomial with one variable since it only contains the variable |\color{green}{x}.|
Consider the following expression. ||\color{green}{x}\color{red}{y}^3+\color{green}{x}\color{blue}{z}^2-3\color{green}{x}+6d|| It is a multivariable polynomial since it contains four different variables, namely |\color{green}{x},\color{red}{y},\color{blue}{z},| and |d.|
There are two special types of polynomials. When a polynomial is made up of two terms, it is called a binomial, and when it has three terms, it is called a trinomial. If there are four or more terms, the expression is simply called a polynomial.
Binomials are algebraic expressions containing two terms. A binomial is in fact the algebraic sum or difference of two monomials.
|\color{green}{6xy^2z^3}+\color{red}{4}| and |\color{green}{34d}-\color{red}{8z}| are binomials since they contain two terms connected by the symbols |+| and |-|.
Trinomials are algebraic expressions containing three terms. A trinomial is in fact the algebraic sum or difference of three monomials.
|\color{green}{6xy^2z^3}-\color{red}{34d}+\color{blue}{5}| is a trinomial since it contains three terms connected by the symbols |+| and |-|.
The degree of an algebraic expression corresponds to the value of the variables’ exponents. Finding the degree varies depending on whether it is a monomial or a polynomial.
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Degree of an algebraic expression |
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The degree of a monomial with a single variable corresponds to the exponent on the variable. |
|15| has a degree of |0| because |15 = 15x^0.| |-3m| has a degree of |1| because |-3m = -3m^1.| |x^2| has a degree of |2.| |7y^3| has a degree of |3.| |
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The degree of a multivariable monomial corresponds to the sum of the exponents of the different variables. |
|2ab| is a |2^\text{nd}| degree monomial because |1+1 = 2.| |5xy^2| is a |3^\text{rd}| degree monomial because |1+2 = 3.| |6d^2e^3| is a |5^\text{th}| degree monomial because |2+3 = 5.| |
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The degree of a polynomial is the degree of the monomial with the highest degree. |
|2x+3| is a |1^\text{st}| degree polynomial because |2x| is the monomial with the greatest degree in the polynomial. |7x^2 + y + 15| has a degree of |2| because |7x^2| is the monomial with the greatest degree in the polynomial. |6a^2c^4 + 3b^3 + 12| has a degree of |6| because |6a^2c^4| is the monomial with the greatest degree in the polynomial. |
The exponents in monomials, binomials, trinomials, and polynomials are always natural numbers.
|3x^{1/2}+2x-4| is not a polynomial since the exponent of the variable |x| is not a natural number.
|-3x^{-1}| is not a monomial since the exponent of the variable |x| is not a natural number.