Standard Form of a Polynomial

A polynomial is a monomial or a sum or difference of monomials. In algebraic expressions such as polynomials, we use plus and minus signs to separate individual terms. Therefore, we can conclude that each term of a polynomial is always a monomial.

Every term of a polynomial is always a monomial.

A trinomial is a polynomial containing 3 different monomial terms. Since only like terms can be combined, the difference of two trinomials can have different outcomes. Let's show one where the difference becomes a binomial. (x^3+4+x) - (x^2+x+4) ⇓ x^3-x^2 However, another set of trinomials could give a difference that is a trinomial. (x^3+4+x) - (x^2+x-4) ⇓ x^3 - x^2 + 8 Consequently, the difference of two trinomials is sometimes a trinomial.

The difference of two trinomials is sometimes a trinomial.

As we discussed, a polynomial is either a monomial or a sum or difference of monomials. When we add two polynomials, we combine like terms. The sum or difference of like terms will result in new monomials, which means the sum of two polynomials must always be another polynomial.

The sum of two polynomials is always a polynomial.

Notice that we can sometimes obtain 0 as a result of adding polynomials. However, like any constant value, 0 is considered a constant polynomial. In fact, it is called the zero polynomial.

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.

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.	$x^{3} - 9 x + 4$

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

Standard Form of a Polynomial

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

Conditions for a

Conditions for n

Standard Form of a Polynomial

Recommended exercises

	12 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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
$4 x + x^{\frac{1}{2}}$	$2$
$2 x^{2} + 5 + y^{- 2}$	$3$
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	$4$

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

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

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