Understanding Polynomial Standard Form: Monomial, Degree, and More

Expression	Why It Is a Monomial
$5$	Any constant is a valid monomial. By the Zero Exponent Property, $5 x^{0} = 5 .$
$0$	The coefficient of a monomial can be $0 .$
$- 2 x^{5}$	The coefficient can be negative.
$\frac{x ^{3} y}{5}$	A monomial can have numbers in the denominator.

Expression

Why It Is a Monomial

5

Any constant is a valid monomial. By the Zero Exponent Property,

5 x^{0} = 5 .

0

The coefficient of a monomial can be

0 .

- 2 x^{5}

The coefficient can be negative.

\frac{x ^{3} y}{5}

A monomial can have numbers in the denominator.

Expression	Why It Is Not a Monomial
$2 x^{- 1}$	The variables of a monomial cannot have negative integer exponents.
$4 \frac{x ^{3}}{y}$	Monomials cannot have variables in the denominator.
$5 x^{3} y^{\frac{1}{2}}$	The variables of a monomial must only have whole number exponents.

Expression

Why It Is Not a Monomial

2 x^{- 1}

The variables of a monomial cannot have negative integer exponents.

4 \frac{x ^{3}}{y}

Monomials cannot have variables in the denominator.

5 x^{3} y^{\frac{1}{2}}

The variables of a monomial must only have whole number exponents.

Monomial	Degree
$3 x$	$1$
$x^{2}$	$2$
$9 x^{3}$	$3$
$x^{3} y$	$4$
$7$	$0$
$\frac{a ^{3} b ^{4} c ^{5}}{1 3}$	$12$
$0$	undefined

Monomial

Degree

3 x

1

x^{2}

2

9 x^{3}

3

x^{3} y

4

7

0

\frac{a ^{3} b ^{4} c ^{5}}{1 3}

12

0

undefined

Name	Definition	Example
Monomial	A polynomial with a single term.	$3 x^{2} y^{3}$
Binomial	A polynomial with a exactly two terms.	$5 x y + 3 x^{2} y^{3}$
Trinomial	A polynomial with exactly three terms.	$x^{3} - 9 x + 4$

Name

Definition

Example

Monomial

A polynomial with a single term.

3 x^{2} y^{3}

Binomial

A polynomial with a exactly two terms.

5 x y + 3 x^{2} y^{3}

Trinomial

A polynomial with exactly three terms.

x^{3} - 9 x + 4

Polynomial	Degree
$- 5 x^{2} y + x^{2} y^{4} - 11 x^{2} - 3$	$6$
$x - 11 x^{4} + 8 x^{3}$	$4$
$7$	$0$
$0$	Undefined

Polynomial

Degree

- 5 x^{2} y + x^{2} y^{4} - 11 x^{2} - 3

6

x - 11 x^{4} + 8 x^{3}

4

7

0

0

Undefined

Expression	Reason
$4 x + x^{\frac{1}{2}}$	It has a non-integer exponent.
$2 x^{2} + 5 + y^{- 2}$	It has a negative exponent.
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	It has a variable in the denominator.

Expression

Reason

4 x + x^{\frac{1}{2}}

It has a non-integer exponent.

2 x^{2} + 5 + y^{- 2}

It has a negative exponent.

\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9

It has a variable in the denominator.

Polynomial	Number of terms
$\frac{x}{4} - 3 x^{3}$	$2$
$2 x^{2} + 5 - 7 y$	$3$
$\frac{x y}{8} + 3 x^{3} - 3 y^{3} + 2 x$	$4$
Expression	Number of terms
$4 x + x^{\frac{1}{2}}$	$2$
$2 x^{2} + 5 + y^{- 2}$	$3$
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	$4$

Polynomial

Number of terms

\frac{x}{4} - 3 x^{3}

2

2 x^{2} + 5 - 7 y

3

\frac{x y}{8} + 3 x^{3} - 3 y^{3} + 2 x

4

Expression

Number of terms

4 x + x^{\frac{1}{2}}

2

2 x^{2} + 5 + y^{- 2}

3

\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9

4

a To simplify the polynomial, like terms will be identified and grouped.

x^{2} + 4 x^{3} - 5 x^{3} + 6 x^{2} + 2 x^{3} ⇕ x^{2} + 6 x^{2} + 4 x^{3} - 5 x^{3} + 2 x^{3}

Now, like terms will be combined by adding or subtracting their coefficients.

x^{2} + 6 x^{2} + 4 x^{3} - 5 x^{3} + 2 x^{3}

FactorOut

Factor out $x^{2} & x^{3}$

(1 + 6) x^{2} + (4 - 5 + 2) x^{3}

AddSubTerms

Add and subtract terms

7 x^{2} + x^{3}

Note that this polynomial has exactly two terms. Therefore, it is a binomial. Next, each term will be analyzed to determine its degree and coefficient.

Term	Degree	Coefficient
$7 x^{2}$	$2$	$7$
$x^{3} \Leftrightarrow (1) x^{3}$	$3$	$1$

To write the polynomial in standard form, its terms will be arranged according to their degree, in descending order.

Standard Form x^{3} + 7 x^{2}

Finally, recall that the degree and leading coefficient of a polynomial are the same as those of the monomial of highest degree. Since the highest degree monomial is

(1) x^{3},

the polynomial's degree is

3

and its leading coefficient is

1 .

The following table summarizes all the information found.

Standard Form
$x^{3} + 7 x^{2}$
Classification	Degree	Leading Coefficient
Binomial	$3$	$1$

b The same steps as in Part A will be followed. The polynomial will be simplified by identifying and combining like terms.

8 x + 4 x^{3} - 2 x^{5} + 7 x - 12 x^{3}

CommutativePropAdd

Commutative Property of Addition

8 x + 7 x + 4 x^{3} - 12 x^{3} - 2 x^{5}

FactorOut

Factor out $x & x^{3}$

(8 + 7) x + (4 - 12) x^{3} - 2 x^{5}

AddSubTerms

Add and subtract terms

15 x - 8 x^{3} - 2 x^{5}

Note that this polynomial has exactly three terms. Therefore, it is a trinomial. Now, each term's degree and coefficient will be identified.

Term	Degree	Coefficient
$15 x \Leftrightarrow 15 x^{1}$	$1$	$15$
$- 8 x^{3}$	$3$	$- 8$
$- 2 x^{5}$	$5$	$- 2$

Next, the polynomial will be written in standard form sorting its terms by degree in descending order.

Standard Form - 2 x^{5} - 8 x^{3} + 15 x

Finally, since the highest degree monomial is

- 2 x^{5},

the polynomial's degree is

5

and its leading coefficient is

- 2 .

The information found will be summarized using a table.

Standard Form
$- 2 x^{5} - 8 x^{3} + 15 x$
Classification	Degree	Leading Coefficient
Trinomial	$5$	$- 2$

Polynomial Equation	Degree
$2 x^{3} + 8 x^{2} + 2 x - 12 = 0$	$3$
Solutions
$x = - 3, x = - 2, x = 1$

Polynomial Equation

Degree

2 x^{3} + 8 x^{2} + 2 x - 12 = 0

3

Solutions

x = - 3, x = - 2, x = 1

{{ article.displayTitle }}

Catch-Up and Review

Classifying Algebraic Expressions

Defining Monomials

Degree - Monomial

Identifying Monomials

Calculating the Degree of a Monomial

Defining Polynomials

Classifying Polynomials by Their Number of Terms

Degree - Polynomial

Leading Coefficient

Identifying Polynomials

Identifying a Polynomial's Degree and Leading Coefficient

Identifying Polynomials and Their Number of Terms

Hint

Solution

Simplifying and Writing Polynomials in Standard Form

Standard Form - Polynomial

Like Terms

Combining Like Terms

Simplifying Polynomials and Identifying Their Characteristics

Answer

Hint

Solution

Introduction to Polynomial Equations

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}