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 3x-8+4x^5 \qquad -7a^2+9b-4b^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—“five-term 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*} has an exponent of 2, so it is a term to the second degree.

\begin{align*}-2x^5\end{align*} has an exponent of 5, so it is a term to the fifth degree.

\begin{align*}x^2y\end{align*} has an exponent of 2 on the \begin{align*}x\end{align*} and an unwritten exponent of 1 on the \begin{align*}y\end{align*},

so this term is to the third degree \begin{align*}(2 + 1)\end{align*}. Notice that we add the two degrees together because it has two variables.

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*}3x-8+4x^5\end{align*} in standard form.

This is a trinomial. \begin{align*}3x\end{align*} has a degree of 1, -8 has a degree of zero, and \begin{align*}4x^5\end{align*} has a degree of 5. We write these in order by degree, highest to lowest:

\begin{align*}4x^5+3x-8\end{align*}

The degree of a polynomial is the same as the degree of the highest term, so this expression is called “a fifth-degree 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^4-2xy+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^3-2xy+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+x-1\end{align*}
- \begin{align*}9+3y^2-2y\end{align*}
- \begin{align*}8+3y^3+8y+9y^2\end{align*}
- \begin{align*}y+6y^4-2y^3+y^2\end{align*}
- \begin{align*}-16y^6-18\end{align*}
- \begin{align*}3x+2x^2+9y+8\end{align*}
- \begin{align*}8y^4+y-7y^3-3y^2\end{align*}
- \begin{align*}-3+8x^2-2x^3-x\end{align*}
- \begin{align*}9-3y^2-2y^3+2y\end{align*}
- \begin{align*}14+6x^2-2x-8y\end{align*}
- \begin{align*}4x+3x^2-5x^3+8x^4\end{align*}
- \begin{align*}-8+3y^2-2y^3+y\end{align*}
- \begin{align*}9+8y^2+2y^3-8y\end{align*}
- \begin{align*}m^4-12m^7+6m^5-6m-8\end{align*}
- \begin{align*}-x^3y^2+5x^3y+8xy\end{align*}