2.1: The Anatomy of a Polynomial

Polynomials are a class of functions that are studied in calculus because they are predictable in their behavior. They are smooth, continuous curves when graphed. They are fairly easy to graph, find roots, and calculate outputs for real-number inputs. Polynomials will continue to be relevant to all of your math and science courses going forward!

Let’s start with the prefixes in the following words:

• Monomial: Mono means “single” or “one.”
• Binomial: Bi means “two.”
• Trinomial: Tri means “three.”
• Polynomial: Poly means “many” or “multi” or “one or more.”

Definition: Monomial

A monomial is a single-term expression in which real numbers multiply variables with whole-number exponents.

The following expressions are examples of monomials:

\(7x^3\;\;\;\;\;\; \dfrac{1}{2}xy\;\;\;\;\;\; 22\;\;\;\;\;\; pq^5\;\;\;\;\;\; \pi r^2\;\;\;\;\;\; −10a^4bc\)

Whole numbers are the counting numbers, starting with zero: \(0\), \(1\), \(2\), \(3\), \(4\), …

The following expressions are not monomials because the exponents are not whole numbers.

\(\begin{array} &&\textcolor{red}{\times}\;\;\;\;\; 2x^{−1} &\text{The exponent on \(x\) is \(–1\).} \\ &\textcolor{red}{\times}\;\;\;\;\; 4u^{\pi}v &\text{The exponent on \(u\) is \(\pi\).} \\ &\textcolor{red}{\times}\;\;\;\;\; 5\sqrt{y} &\text{The exponent on \(y\) is \(\dfrac{1}{2}\).} \\ &\textcolor{red}{\times}\;\;\;\;\; \dfrac{3}{4t^2} &\text{The exponent on \(t\) is \(–2\).} \end{array}\)

Definition: Binomial

A binomial is a two-term expression in which two monomials are added or subtracted to form a single expression of 2 terms.

The following expressions are examples of binomials:

\(3x + 1\;\;\;\;\;\; x^4 − y^4\;\;\;\;\;\; 5y^5 − 5y \;\;\;\;\;\; \pi r^2 + 2 \pi r h\)

Definition: Trinomial

A trinomial is a three-term expression in which three monomials are added or subtracted.

The following expressions are examples of trinomials:

\(x^3 + 4x^2 − 3 \;\;\;\;\;\; p^2q^2 − 5pq + 6 \;\;\;\;\;\; \dfrac{1}{4}n^2 − mn − \dfrac{3}{2}n \;\;\;\;\;\; 6t^{10} + 2 \pi t^2 + \pi\)

Definition: Polynomial

A polynomial is an expression consisting of one or more terms, and each term is a monomial.

Monomials, binomials, and trinomials are special names of polynomials with \(1\), \(2\), or \(3\) terms. You could call a trinomial a polynomial and that is fine! Once the expression extends beyond 3 terms, the polynomial has no special name; it is just a polynomial!

Definition: Degree of a Polynomial

To find the degree of a polynomial, inspect each term’s exponents. Each term of the polynomial has its own degree. A term’s degree is found by summing the exponents of the term. The term with the highest exponent-sum becomes the degree of the polynomial.

Example 2.1.1

Find the degree of each polynomial.

1. \(p^2q^2 − 5pq + 6\)
2. \(2y^5 − 4x^4y^3 + 10y^6 − y\)

Solution

a) The degree of the trinomial is determined by the term of highest degree

\(\underbrace{p^2q^2}_{\text{Term} 1} - \underbrace{5pq}_{\text{Term} 2} + \underbrace{6}_{\text{Term} 3} \)

Term 1	\(p^2q^2\)	Degree \(= 2 + 2 = 4\) \(\textcolor{green}{\checkmark}\)
Term 2	\(−5p^1q^1\)	Degree \(= 1 + 1 = 2\)
Term 3	\(6p^0q^0\)	Degree \(= 0 + 0 = 0\)

Answer The degree of the trinomial is \(4\).

b) Determine the degree of each of the \(4\) terms. The term of highest degree is the degree of the polynomial.

\(\underbrace{2y^5}_{\text{Term} 1} - \underbrace{4x^4y^3}_{\text{Term} 2} + \underbrace{10y^6}_{\text{Term} 3} - \underbrace{y}_{\text{Term} 4} \)

Term 1	\(2y^5\)	Degree \(= 5\)
Term 2	\(4x^4y^3\)	Degree \(= 4 + 3 = 7\) \(\textcolor{green}{\checkmark}\)
Term 3	\(10y^6\)	Degree \(= 6\)
Term 4	\(y\)	Degree \(=1\)

Answer The degree of the trinomial is \(7\).

Associated Vocabulary

The terms \(p^2q^2\) and \(−5pq\) are variable terms, and the term “\(6\)” is called a constant term. That is, a term without variables is a constant term. Each variable term has a numerical factor which is called a coefficient of the term. A polynomial often has terms stated in the descending order of degree. Therefore, the term of highest degree is often stated first, and this is why the term of highest degree is called the leading term of the polynomial. The coefficient of the leading term is called the leading coefficient.

The table below summarizes polynomial vocabulary and key concepts:

	Polynomial	Descending Order	Leading Coefficient	Degree of Polynomial
Monomials	\(-8x^2y\)	\(-8x^2y\)	\(-8\)	\(3\)
	\(15\)	\(15\)	\(15\)	\(0\)
Binomials	\(−6n^2m^3 + 3n^7\)	\(3n^7 − 6n^2m^3\)	\(3\)	\(7\)
	\(\dfrac{2}{3} - \dfrac{1}{3}b\)	\(-\dfrac{1}{3}b + \dfrac{2}{3}\)	\(-\dfrac{1}{3}\)	\(1\)
Trinomials	\(3-5c-c^2\)	\(-c^2-5c+3\)	\(-1\)	\(2\)
	\(2 \pi r^3h − r^2 + \pi r^3h^2\)	\(\pi r^3h^2 + 2 \pi r^3h − r^2\)	\(\pi\)	\(5\)

A Single Variable Polynomial and the Function p(x)

Most of your work will be with polynomials of a single variable. The following is a formal definition of a single variable polynomial function, \(p(x)\).

Definition: Polynomial Function

A polynomial function \(p(x)\) is a sum of the terms \(a_nx^n\) where \(a_0, a_1, a_2,...,a_n\) are real numbers and \(n\) is a nonnegative integer.

\[p(x) =a_0, a_1x, a_2x^2,...,a_nx^n\]

The above function notation may seem unnecessarily complicated at first glance. However, notice that if we used the alphabet letters A through Z for coefficients, we are limited to \(26\) terms. Therefore, by using the letter \(a\) with a numerical subscript rather than using letters A-Z, we don’t have any limitation on the number of terms.

Try It! (Exercises)

For #1-9, say whether each expression is a polynomial. If it is, identify it as a monomial, binomial, or trinomial.

1. \(12t^4 s^2\)
2. \(1 − \dfrac{1}{6}p\)
3. \(\dfrac{4}{c} − 1\)
4. \(3y^4 − 2x^2\)
5. \(4v − u^2 + uv\)
6. \(8r^{\pi}\)
7. \(−3.8 − 6.2x + 0.4x^2\)
8. \(6 \pi r 2d\)
9. \(5a^2 − 4a + a −1\)
10. Create your own example of a monomial of degree eight.
11. Create your own example of a trinomial of degree five.
12. Create your own example of a binomial of degree four.
13. The degree of a nonzero constant term such as \(3\) is zero. Explain why.

For #14-19, fill in the table appropriately for each polynomial.

	Polynomial	Leading Term	Leading Coefficient	Degree
14.	\(1-x^3\)
15.	\(−6a^5b^3\)
16.	\(3p^5q^4 − 5p^5 + 6p^6q\)
17.	\(2 + \dfrac{3}{4}x\)
18.	\(5\pi − y − 7y^3 + 4 \pi y^4 − y^2\)
19.	\(1 + 0.25xy + 0.65y\)

20. Fill in the blank with the correct vocabulary.

1. A ____________ is a polynomial of a single term.
2. For the term \(bx^n\), \(b\) is called ____________ the of the term and \(n\) is the ____________ of the term.
3. The term with the highest degree is called the ____________ term.
4. All linear functions of the form \(f(x) = mx + b\) are polynomials that have degree \(=\) ____________.