### Polynomials in Standard Form

So far we’ve seen functions described by straight lines (linear functions) and functions where the variable appeared in the exponent (exponential functions). In this section we’ll introduce polynomial functions. A **polynomial** is made up of different terms that contain **positive integer** powers of the variables. Here is an example of a polynomial:

\begin{align*}4x^3+2x^2-3x+1\end{align*}

Each part of the polynomial that is added or subtracted is called a **term** of the polynomial. The example above is a polynomial with *four terms*.

The numbers appearing in each term in front of the variable are called the **coefficients**. The number appearing all by itself without a variable is called a **constant**.

In this case the coefficient of \begin{align*}x^3\end{align*} is **4**, the coefficient of \begin{align*}x^2\end{align*} is **2**, the coefficient of \begin{align*}x\end{align*} is **-3** and the constant is **1**.

**Degrees of Polynomials and Standard Form**

Each term in the polynomial has a different **degree**. The degree of the term is the power of the variable in that term.

\begin{align*}& 4x^3 && \text{has degree} \ 3 \ \text{and is called a cubic term or} \ 3^{rd} \ \text{order term}.\\ & 2x^2 && \text{has degree} \ 2 \ \text{and is called a quadratic term or} \ 2^{nd} \ \text{order term}.\\ & -3x && \text{has degree} \ 1 \ \text{and is called a linear term or} \ 1^{st} \ \text{order term}.\\ & 1 && \text{has degree} \ 0 \ \text{and is called the constant}.\end{align*}

By definition, **the degree of the polynomial** is the same as the degree of the term with the highest degree. This example is a polynomial of degree 3, which is also called a “cubic” polynomial. (Why do you think it is called a cubic?).

Polynomials can have more than one variable. Here is another example of a polynomial:

\begin{align*}t^4-6s^3t^2-12st+4s^4-5\end{align*}

This is a polynomial because all the exponents on the variables are positive integers. This polynomial has five terms. Let’s look at each term more closely.

**Note:** *The degree of a term is the sum of the powers on each variable in the term.* In other words, the degree of each term is the number of variables that are multiplied together in that term, whether those variables are the same or different.

\begin{align*}& t^4 && \text{has a degree of} \ 4, \ \text{so it's a} \ 4^{th} \ \text{order term}\\ & -6s^3t^2 && \text{has a degree of} \ 5, \ \text{so it's a} \ 5^{th} \ \text{order term}.\\ & -12st && \text{has a degree of} \ 2, \ \text{so it's a} \ 2^{nd} \ \text{order term}.\\ & 4s^4 && \text{has a degree of} \ 4, \ \text{so it's a} \ 4^{th} \ \text{order term}.\\ & -5 && \text{is a constant, so its degree is} \ 0.\end{align*}

Since the highest degree of a term in this polynomial is 5, then this is polynomial of degree \begin{align*}5^{th}\end{align*} or a \begin{align*}5^{th}\end{align*} order polynomial.

A polynomial that has only one term has a special name. It is called a **monomial** (*mono* means one). A monomial can be a constant, a variable, or a product of a constant and one or more variables. You can see that each term in a polynomial is a monomial, so a polynomial is just the sum of several monomials. Here are some examples of monomials:

\begin{align*}b^2 \qquad -2ab^2 \qquad 8 \qquad \frac{1}{4}x^4 \qquad -29xy\end{align*}

#### Identifying Constants and the Degree of a Polynomial

For the following polynomials, identify the coefficient of each term, the constant, the degree of each term and the degree of the polynomial.

a) \begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}

\begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}

The coefficients of each term in order are 1, -3, 4, and -5 and the constant is 7.

The degrees of each term are 5, 3, 2, 1, and 0. Therefore the degree of the polynomial is 5.

b) \begin{align*}x^4-3x^3y^2+8x-12\end{align*}

\begin{align*}x^4-3x^3y^2+8x-12\end{align*}

The coefficients of each term in order are 1, -3, and 8 and the constant is -12.

The degrees of each term are 4, 5, 1, and 0. Therefore the degree of the polynomial is 5.

#### Identifying Polynomials

Identify the following expressions as polynomials or non-polynomials.

a) \begin{align*}5x^5-2x\end{align*}

This ** is** a polynomial.

b) \begin{align*}3x^2-2x^{-2}\end{align*}

This is ** not** a polynomial because it has a negative exponent.

c) \begin{align*}x\sqrt{x}-1\end{align*}

This is ** not** a polynomial because it has a radical.

d) \begin{align*}\frac{5}{x^3+1}\end{align*}

This is ** not** a polynomial because the power of \begin{align*}x\end{align*} appears in the denominator of a fraction (and there is no way to rewrite it so that it does not).

e) \begin{align*}4x^\frac{1}{3}\end{align*}

This is ** not** a polynomial because it has a fractional exponent.

f) \begin{align*}4xy^2-2x^2y-3+y^3-3x^3\end{align*}

This ** is** a polynomial.

Often, we arrange the terms in a polynomial in order of decreasing power. This is called **standard form**.

The following polynomials are in standard form:

\begin{align*}4x^4-3x^3+2x^2-x+1\end{align*}

\begin{align*}a^4b^3-2a^3b^3+3a^4b-5ab^2+2\end{align*}

The first term of a polynomial in standard form is called the **leading term**, and the coefficient of the leading term is called the **leading coefficient**.

The first polynomial above has the leading term \begin{align*}4x^4,\end{align*} and the leading coefficient is 4.

The second polynomial above has the leading term \begin{align*}a^4b^3,\end{align*} and the leading coefficient is 1.

#### Writing Polynomials in Standard Form

Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial.

a) \begin{align*}7-3x^3+4x\end{align*}

\begin{align*}7-3x^3+4x\end{align*} becomes \begin{align*}-3x^3+4x+7\end{align*}. Leading term is \begin{align*}-3x^3\end{align*}; leading coefficient is -3.

b) \begin{align*}ab-a^3+2b\end{align*}

\begin{align*}ab-a^3+2b\end{align*} becomes \begin{align*}-a^3+ab+2b\end{align*}. Leading term is \begin{align*}-a^3\end{align*}; leading coefficient is -1.

c) \begin{align*}-4b+4+b^2\end{align*}

\begin{align*}-4b+4+b^2\end{align*} becomes \begin{align*}b^2-4b+4\end{align*}. Leading term is \begin{align*}b^2\end{align*}; leading coefficient is 1.

**Simplifying Polynomials**

A polynomial is simplified if it has no terms that are alike. **Like terms** are terms in the polynomial that have the same variable(s) with the same exponents, whether they have the same or different coefficients.

For example, \begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms, but \begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.

When a polynomial has like terms, we can simplify it by combining those terms.

\begin{align*}& x^2+\underline{6xy} - \underline{4xy} + y^2\\ & \qquad \nearrow \qquad \nwarrow\\ & \qquad \text{Like terms}\end{align*}

We can simplify this polynomial by combining the like terms \begin{align*}6xy\end{align*} and \begin{align*}-4xy\end{align*} into \begin{align*}(6-4)xy\end{align*}, or \begin{align*}2xy\end{align*}. The new polynomial is \begin{align*}x^2+2xy+y^2\end{align*}.

Simplify the following polynomials by collecting like terms and combining them.

a) \begin{align*}2x -4x^2+6+x^2-4+4x\end{align*}

Rearrange the terms so that like terms are grouped together: \begin{align*}(-4x^2+x^2)+(2x+4x)+(6-4)\end{align*}

Combine each set of like terms: \begin{align*}-3x^2+6x+2\end{align*}

b) \begin{align*}2x -4x^2+6+x^2-4+4x\end{align*}

Rearrange the terms so that like terms are grouped together: \begin{align*}(a^3b^3-a^3b^3)+(-5ab^4+3ab^4)+2a^3b-a^2b\end{align*}

Combine each set of like terms: \begin{align*}0-2ab^4+2a^3b-a^2b=-2ab^4+2a^3b-a^2b\end{align*}

### Example

#### Example 1

Simplify and rewrite the following polynomial in standard form. State the degree of the polynomial.

\begin{align*}16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}

Start by simplifying by combining like terms:

\begin{align*} 16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}

is equal to

\begin{align*}(16x^2y^3-7x^2y^3)-3xy^5+(-2x^3y^2+2x^3y^2)+2xy\end{align*}

which simplifies to

\begin{align*}9x^2y^3-3xy^5+2xy.\end{align*}

In order to rewrite in standard form, we need to determine the degree of each term. The first term has a degree of \begin{align*}2+3=5\end{align*}, the second term has a degree of \begin{align*}1+5=6\end{align*}, and the last term has a degree of \begin{align*}1+1=2\end{align*}. We will rewrite the terms in order from largest degree to smallest degree:

\begin{align*}-3xy^5+9x^2y^3+2xy\end{align*}

The degree of a polynomial is the largest degree of all the terms. In this case that is 6.

### Review

Indicate whether each expression is a polynomial.

- \begin{align*}x^2+3x^{\frac{1}{2}}\end{align*}
- \begin{align*}\frac{1}{3}x^2y-9y^2\end{align*}
- \begin{align*}3x^{-3}\end{align*}
- \begin{align*}\frac{2}{3}t^2-\frac{1}{t^2}\end{align*}
- \begin{align*}\sqrt{x}-2x\end{align*}
- \begin{align*}\left ( x^\frac{3}{2} \right )^2\end{align*}

Express each polynomial in standard form. Give the degree of each polynomial.

- \begin{align*}3-2x\end{align*}
- \begin{align*}8-4x+3x^3\end{align*}
- \begin{align*}-5+2x-5x^2+8x^3\end{align*}
- \begin{align*}x^2-9x^4+12\end{align*}
- \begin{align*}5x+2x^2-3x\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.1.