Polynomial components

Understanding Polynomials

How do you tell when you're working with polynomials? And how do you explain different components in a polynomial?

The polynomial definition is quite simple. The prefix "Poly" means "many" and polynomials are sums of variables and exponents.

You can divide up a polynomial into "terms", separated by each part that is being added. Polynomial terms do not have square roots of variables, factional powers, nor does it have variables in the denominator of any fractions it may have. The polynomial terms can only have variables with exponents that are whole-numbers.

In general, polynomials are written with its terms being ordered in decreasing order of exponents. The term with the largest exponent goes first, followed by the term with the next highest exponent and so on till you reach a constant term.

Although polynomials can range from one to a very large amount of terms, you may hear specific names referencing to polynomials of a set number of terms. They are as follows:

- Monomial: A one-term polynomial (e.g. $3x$)

- Bionomial: A two-term polynomial (e.g. $x^4 + 3x$)

- Trinomial: A three-term polynomial (e.g. $x^4 + 2x^2 + 3x$)

If you see the above three names being used in a question, don't worry. It's really just another, more specific, word for polynomials.

Let's learn how to describe the different parts of polynomials through an example.

Question:

Describe the following polynomial: $2x^5 y^2-3xy^2-7$

Solution:

Terms: $2x^5 y^2$, $-3xy^2$, $-7$

To get polynomial coefficients, take the numbers in front of each terms.

Coefficients: $2, -3, -7$

Leading coefficient is the coefficients of the leading term

Leading coefficient: $2$

The get the degree of each terms, add the exponents in each terms to get their degrees.

Degree: $7, 3, 0$

Leading term is the term that has highest degree

Leading term: $2x^5 y^2$

Degree of polynomial is the degree of the leading term.

Degree of polynomial: 7

Constant: $- 7$

How to find polynomial terms

Let's look into this one-by-one. The idea of "terms" is explained in the previous section above, where we explored what is and isn't a polynomial term. To recap, polynomial terms do not have square roots of variables, factional powers, nor does it have variables in the denominator of any fractions it may have. The polynomial terms can only have variables with exponents that are whole-numbers.

How to find polynomial coefficients

Polynomial coefficients are the numbers that come before a term. Terms usually have a number and a variable (e.g. $2x^2$, where $2$ is the number, and $x$ is the variable). The number portion is the coefficient. So what is a leading coefficient? Simply stated, it's the coefficient of the leading term. Since polynomials are sorted based on descending powers of exponents, the first term's (after a polynomial's terms are sorted properly) coefficient is the leading coefficient.

How to find polynomial degree

The degrees are the exponents of terms. When you have multiple exponents in a term, such as in this example, you'll have to add together their exponents to find the total exponents of that term. Keep in mind that if the variable has no exponent, it actually has an exponent of 1. It the term does not have variable, it is degree 0.

How to find degree of polynomial

The leading term is the term that has the highest polynomial degree. In our case, since the exponents of "$5$" and "$2$" add together to get $7$, it has a higher degree than any of the other polynomial terms. Therefore, the leading term is the whole of: $2x^5 y^2$. The leading term helps us find the degree of polynomial. It is simply the degree of the leading term.

How to find the constant

The last component is called the constant. The constant does not have a variable in it, and therefore remains constant no matter what the value of $x$ is when we are trying to evaluate the polynomial equation.