Chapter Eleven — Polynomials

The Humongous Book of Algebra Problems

238

Classifying Polynomials

Labeling them based on the exponent and total terms

11.1 Classify the polynomial 7x + 10 according to its degree and the number of

terms it contains.

The degree of a polynomial is the highest exponent to which the variable in that

polynomial is raised. The only variable present in this expression is x, so the

polynomial has degree one. Such polynomials are described as linear.

The polynomial contains two terms, 7x and 10, so it is a linear binomial.

11.2 Classify the polynomial 6x

2

according to its degree and the number of terms it

contains.

The polynomial 6x

2

consists of a single term, so it is a monomial. That term

contains a variable raised to the second power, which classiﬁes the polynomial

as a quadratic monomial.

11.3 Classify the polynomial –3x

2

+ 5x according to its degree and the number of

terms it contains.

The polynomial –3x

2

contains two terms: –3x

2

and 5x. Of those terms, the

highest power of x is 2, so the polynomial has degree two. A polynomial with

two terms and degree two is called a quadratic binomial.

11.4 Classify the polynomial 12x

3

according to its degree and the number of terms

it contains.

This polynomial consists of a single term, so it is a monomial. Furthermore, the

polynomial has degree three, which means it is cubic. In conclusion, 12x

3

is a

cubic monomial.

Note: Problems 11.5–11.6 refer to the polynomial 4x

2

– 7x + 6.

11.5 Classify the polynomial according to its degree and the number of terms it

contains.

This polynomial contains three terms (4x

3

, –7x, and 6) and has degree two

(because the highest power of x is two). Therefore, 4x

2

– 7x + 6 is a quadratic

trinomial.

It doesn’t

have degree

one because

there’s only one

variable. It has

degree one because

the only variable

there is raised

to the rst

POWER.

It’s a BInomial

because it has

TWO terms (bi =

two). A MONOmial

has ONE term, and

a TRInomial has

THREE terms.

RULE

OF THUMB:

The degree is

the highest power

of the variable.

Here’s how polynomial

classications break

down by degree:

Degree 1: Linear

Degree 2: Quadratic

Degree 3: Cubic

Degree 4: Quartic

Degree 5: Quintic

There are others, but

these are the most

common.

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