12.2: Write and Classify Polynomials in Standard Form
Do you know how to measure the degree of a polynomial? Take a look at this dilemma.
\begin{align*}4x^3+3x+9\end{align*}
What if you had this polynomial? Can you identify it by degree? Is it in standard form?
This Concept will teach you all that you need to know to answer these questions.
Guidance
Sometimes, you will see an expression or an equation that has exponents and variables in it. These expressions and equations can have more than one variable and sometimes more than one exponent in them. To understand how to work with these variables and exponents, we have to understand polynomials.
A polynomial is an algebraic expression that shows the sum of monomials.
Polynomials: \begin{align*}x^2+ 5 \qquad 3x8+4x^5 \qquad 7a^2+9b4b^3+6\end{align*}
We call an expression with a single term a monomial, an expression with two terms is a binomial, and an expression with three terms is a trinomial. An expression with more than three terms is named simply by its number of terms—“fiveterm polynomial.”
Did you know that you can write and classify polynomials?
First, let’s think about how we can classify each polynomial. We classify them according to terms. Each term can be classified by its degree.
The degree of a term is determined by the exponent of the variable or the sum of the exponents of the variables in that term.
\begin{align*}x^2\end{align*}
\begin{align*}2x^5\end{align*}
\begin{align*}x^2y\end{align*}
so this term is to the third degree \begin{align*}(2 + 1)\end{align*}
8 is a monomial that is a constant with no variable, its degree is zero.
We can also work on the ways that we write polynomials. One way to write a polynomial is in what we call standard form.
In order to write any polynomial in standard form, we look at the degree of each term. We then write each term in order of degree, from highest to lowest, left to write.
Take a look.
Write the expression \begin{align*}3x8+4x^5\end{align*}
This is a trinomial. \begin{align*}3x\end{align*}
\begin{align*}4x^5+3x8\end{align*}
The degree of a polynomial is the same as the degree of the highest term, so this expression is called “a fifthdegree trinomial.”
Name the degree of each polynomial.
Example A
\begin{align*}5x^4+3x^3+9x^2\end{align*}
Solution: Fourth degree
Example B
\begin{align*}6y^3+3xy+9\end{align*}
Solution: Third degree
Example C
\begin{align*}7x^2+3x+9y\end{align*}
Solution: Second degree
Now let's go back to the dilemma from the beginning of the Concept.
\begin{align*}4x^3+3x+9\end{align*}
The highest exponent here is a 3, so we can say that this is a polynomial to the third degree. Since the values go from the highest degree to the least degree, we can say that this polynomial is in standard form already.
Vocabulary
 Polynomial
 an algebraic expression that shows the sum of monomials. A polynomial can also be named when there are more than three terms present.
 Monomial
 an expression where there is one term.
 Binomial
 an expression where there are two terms.
 Trinomial
 an expression where there are three terms.
 Constant
 a term that is a single number such as 4 or 9.
 Coefficient
 a variable and a numerical factor and the numerical factor is the coefficient
Guided Practice
Here is one for you to try on your own.
Write the following polynomial in standard form.
\begin{align*}4x^3+3x^5+9x^42xy+11\end{align*}
Solution
To accomplish this task, rewrite the polynomials so that the exponents are in descending order.
\begin{align*}3x^5+9x^4+4x^32xy+11\end{align*}
This is our answer.
Video Review
Practice
Directions: Write the following polynomials in standard form and then identify its degree:

\begin{align*}4x^2+5x^3+x1\end{align*}
4x2+5x3+x−1 
\begin{align*}9+3y^22y\end{align*}
9+3y2−2y 
\begin{align*}8+3y^3+8y+9y^2\end{align*}
8+3y3+8y+9y2 
\begin{align*}y+6y^42y^3+y^2\end{align*}
y+6y4−2y3+y2 
\begin{align*}16y^618\end{align*}
−16y6−18 
\begin{align*}3x+2x^2+9y+8\end{align*}
3x+2x2+9y+8 
\begin{align*}8y^4+y7y^33y^2\end{align*}
8y4+y−7y3−3y2 
\begin{align*}3+8x^22x^3x\end{align*}
−3+8x2−2x3−x 
\begin{align*}93y^22y^3+2y\end{align*}
9−3y2−2y3+2y 
\begin{align*}14+6x^22x8y\end{align*}
14+6x2−2x−8y 
\begin{align*}4x+3x^25x^3+8x^4\end{align*}
4x+3x2−5x3+8x4 
\begin{align*}8+3y^22y^3+y\end{align*}
−8+3y2−2y3+y 
\begin{align*}9+8y^2+2y^38y\end{align*}
9+8y2+2y3−8y 
\begin{align*}m^412m^7+6m^56m8\end{align*}
m4−12m7+6m5−6m−8 
\begin{align*}x^3y^2+5x^3y+8xy\end{align*}
−x3y2+5x3y+8xy
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Binomial
A binomial is an expression with two terms. The prefix 'bi' means 'two'.Coefficient
A coefficient is the number in front of a variable.constant
A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.Monomial
A monomial is an expression made up of only one term.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Polynomial Function
A polynomial function is a function defined by an expression with at least one algebraic term.Trinomial
A trinomial is a mathematical expression with three terms.Image Attributions
Here you'll write and classify polynomials in standard form.