Introduction

Polynomials with one, two, or three terms have special names.

  • A polynomial with one term is called a monomial. For example: 1, \qquad y, \qquad 2x^3, \qquad 5a^{4} However, the following expressions are not monomials because they can't be called polynomials: \dfrac{1}{y}, \qquad 5t^{-1}, \qquad \sqrt{x}, \qquad 2z^{1/4}

  • A polynomial with two terms is called a binomial. For example: x+3, \qquad 7n^2 +n, \qquad p^{32} -1

  • A polynomial with three terms is called a trinomial. For example: z^2 -z +1, \qquad x^3 -2x -1, \qquad 2c^{4} - 3c^{3} +4c^2

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Example: Categorizing a Polynomial by the Number of Terms It Has

Categorize the polynomial 2y^2+3 based on the number of terms it has.

EXPLANATION

The polynomial 2y^2+3 has two terms, so it is a binomial.

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Practice: Categorizing a Polynomial by the Number of Terms It Has

Which of the following expressions is a monomial?

a
 $7z$
b
 $3z+1$
c
 $z^4-3z-2$
d
 $z^2+z$
e
 $z^2-2$
Practice: Categorizing a Polynomial by the Number of Terms It Has

Which of the following statements are true?

  1. $-\dfrac {v^5}{10}$ is a binomial
  2. $\dfrac {u^7}{5}$ is a monomial
  3. $7.5y^2+9.5$ is a trinomial
a
 II only
b
 III only
c
 II and III only
d
 I and III only
e
 I and II only
Names of Polynomials Based on Their Degree

Each type of polynomial of degree between 0 and 5 has a specific name. These are shown in the table below.


Degree Name Example
0 Constant 7
1 Linear 3x + 2
2 Quadratic 2x^2 - 6x + 1
3 Cubic x^3 + 4x^2 + 9x
4 Quartic 3x^4 - 5x^2
5 Quintic 7x^5 + 3x^4 - x^4 + x - 6


We can combine these with the names we learned earlier.

For example:

  • 2x+1 is a linear binomial because its degree equals 1, and it has 2 terms.

  • x^2 is a quadratic monomial because its degree equals 2, and it has 1 term.

  • 2x^3+x+1 is a cubic trinomial because its degree equals 3, and it has 3 terms.

If a polynomial contains more than three terms, then we refer to it by its degree only.

For example:

  • x^3+2x^2+x-1 is a cubic polynomial.
  • x^4+x^3+x^2+x+1 is a quartic polynomial.
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Example: Categorizing a Polynomial by Its Degree and Number of Terms

Categorize the polynomial 1+2x^2 - 8x based on its degree and the number of terms it has.

EXPLANATION

The polynomial 1+2x^2 - 8x has three terms, so it is a trinomial.

Also, its degree is 2, so it is quadratic.

Therefore, the polynomial is a quadratic trinomial.

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Practice: Categorizing a Polynomial by Its Degree and Number of Terms

Categorize the polynomial $2x^3+x$ based on its degree and the number of terms it has.

a
 quadratic trinomial
b
 cubic binomial
c
 linear binomial
d
 quadratic binomial
e
 cubic trinomial
Practice: Categorizing a Polynomial by Its Degree and Number of Terms

Categorize the polynomial $3t^4 - 5t^3 + 2$ based on its degree and the number of terms it has.

a
 cubic binomial
b
 quartic binomial
c
 quadratic monomial
d
 quartic trinomial
e
 cubic trinomial
Example: Identifying True Statements

Which of the following statements are true?

  1. The sum of x^2+2 and 3x is a trinomial.
  2. The sum of 5x^2+4x and -4x is a trinomial.
  3. The sum of a monomial and a binomial is always a trinomial.
EXPLANATION

Let's analyze each statement, in turn.

  • Statement I is true. We have (x^2+2) + (3x) = x^2+3x+2, which is a trinomial.

  • Statement II is false. We have (5x^2+4x) + (-4x) = 5x^2, which is not a trinomial (it's a monomial).

  • Statement III is false. The sum of a monomial and a binomial could be a trinomial (see, for example, the statement I).

    However, the sum of a monomial and a binomial is not always a trinomial (see, for example, the statement II).

    So, the sum of a monomial and a binomial is not always a trinomial.

Therefore, the correct answer is "I only."

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Practice: Identifying True Statements

Which of the following statements are true?

  1. The sum of $2x^3$ and $-x^2$ is a binomial.
  2. The sum of $x^4$ and $3x^4$ is a binomial.
  3. The sum of two monomials is always a binomial.
a
 I, II, and III
b
 II and III only
c
 I and II only
d
 II only
e
 I only
Practice: Identifying True Statements

Which of the following statements are true?

  1. The product of $6x^2$ and $11x$ is a monomial.
  2. The product of $x^3$ and $x^2$ is a monomial.
  3. The product of two monomials is never a monomial.
a
 II only
b
 III only
c
 I, II, and III
d
 I and II only
e
 II and III only
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