Polynomial by definition means many terms. A term is an expression consisting of constants, variables, and exponents. A polynomial consists of terms separated by addition/subtraction.
It is important to note that some expressions, while contains many terms, are in fact not polynomials.
Outlined below are a few rules indicating the characteristics that polynomials cannot have:
The degree of any individual term of a polynomial is determined by the exponent of its variable.For terms that contain more than one variable the degree would be the sum of the exponents on those variables. The degree of a polynomial is simply the degree of the highest term.
The leading coefficient is the coefficient on the term with the highest degree.
In the first term, \( ab^2 \), the degree is \(3\), in the second term, \( a^2b^3 \), the degree is \(5\), and in the third term \( a^3b \) the degree is \(4\). Therefore, we can determine the degree of this polynomial is \(\bolsymbol{5}\) since it's the highest value.
Other than degrees, a polynomial can also be categorized by terms, such that a monomial has one term, a binomial has two terms, and a trinomial with three terms.
Try and fit the following polynomials into their correct description.
\(4 - 5x\)
\(-24m^2\)
\(a^2 + b^2\)
\(3xy^{-2}\)
\(3x^2 - x + 5\)
\(\frac{1}{2}b\)
\(-\frac{1}{x^2}\)
\(a^2 - b - 5\)