Suppose that two square plots of land both have a side length of \begin{align*}x\end{align*}

### Guidance

So far we have discussed linear functions and exponential functions. This Concept introduces polynomial functions.

**Definition:** A **polynomial** is an expression made with constants, variables, and *positive integer* exponents of the variables.

An example of a polynomial is: \begin{align*}4x^3 + 2x^2 - 3x + 1\end{align*}**coefficients.** 4, 2, and -3 are coefficients because those numbers are in front of a variable. The number appearing all by itself without a variable is called a **constant.** 1 is the constant because it is by itself.

#### Example A

*Identify the following expressions as polynomials or non-polynomials.*

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

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

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

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

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

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

**Solution:**

(a) \begin{align*}5x^2 - 2x\end{align*}** is** a polynomial.

(b) \begin{align*}3x^2 - 2x^{-2}\end{align*}** not** a polynomial because it has a negative exponent.

(c) \begin{align*}x\sqrt{x} - 1\end{align*}** not** a polynomial because is has a square root.

(d) \begin{align*}\frac{5}{x^3 + 1}\end{align*}** not** a polynomial because the power of \begin{align*}x\end{align*}

(e) \begin{align*}4x^{\frac{1}{3}}\end{align*}** not** a polynomial because it has a fractional exponent.

(f) \begin{align*}4xy^2 - 2x^y - 3 + y^3 - 3x^3\end{align*}** is** a polynomial.

**Classifying Polynomials by Degree**

The **degree** of a polynomial is the largest exponent of a single term.

- \begin{align*}4x^3\end{align*}
4x3 has a degree of 3 and is called a**cubic term**or \begin{align*}3^{rd}\end{align*}3rd **order term**. - \begin{align*}2x^2\end{align*}
2x2 has a degree of 2 and is called a**quadratic term**or \begin{align*}2^{nd}\end{align*}2nd **order term**. - \begin{align*}-3x\end{align*}
−3x has a degree of 1 and is called a**linear term**or \begin{align*}1^{st}\end{align*}1st **order term.** - 1 has a degree of 0 because there is no variable.

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*}**Note:** *The degree of a term is the sum of the powers on each variable in the term.*

\begin{align*}t^4\end{align*}

\begin{align*}-6s^3t^2\end{align*}

\begin{align*}-12^{st}\end{align*}

\begin{align*}4s^4\end{align*} has a degree of 4, so it’s a \begin{align*}4^{th}\end{align*} order term.

–5 is a constant, so its degree is 0.

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

#### Example B

*Identify the coefficient on each term, the degree of each term, and the degree of the polynomial.*

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

**Solution:** 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.

A **monomial** is a one-termed polynomial. It can be a constant, a variable, or a variable with a coefficient. Examples of monomials are the following: \begin{align*}b^2; \ 6; \ -2ab^2; \ \frac{1}{4} x^2\end{align*}

**Rewriting Polynomials in Standard Form**

Often, we arrange the terms in a polynomial in **standard form** in which the term with the highest degree is first and is followed by the other terms in order of decreasing powers. The first term of a polynomial in this form is called the *leading term*, and the coefficient in this term is called the *leading coefficient*.

#### Example C

*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*}

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

**Solution:**

(a) \begin{align*}7-3x^3+4x\end{align*} is rearranged as \begin{align*}-3x^3+4x+7\end{align*}. The leading term is \begin{align*}-3x^3\end{align*} and the leading coefficient is –3.

(b) \begin{align*}ab-a^3+2b\end{align*} is rearranged as \begin{align*}-a^3+ab+2b\end{align*}. The leading term is \begin{align*}-a^3\end{align*} and the 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, but they can have different coefficients.

\begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms.

\begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.

If we have a polynomial that has like terms, we simplify by combining them.

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

This polynomial is simplified by combining the like terms: \begin{align*}6xy-4xy=2xy\end{align*}. We write the simplified polynomial as \begin{align*}x^2+2xy+y^2\end{align*}.

#### Example D

*Simplify by collecting and combining like terms.*

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

**Solution:** Use the Commutative Property of Addition to reorganize like terms and then simplify.

\begin{align*} = (a^3b^3-a^3b^3) + (-5ab^4+3ab^4) + 2a^3b-a^2b\\ = 0-2ab^4+2a^3b-a^2b\\ = -2ab^4+2a^3 b-a^2 b\end{align*}

### Video Review

### Guided Practice

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*}

**Solution:**

Start by simplifying by combining like terms:

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

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Addition and Subtraction of Polynomials (15:59)

Define the following key terms.

- Polynomial
- Monomial
- Degree
- Leading coefficient

For each of the following expressions, decide whether it is a polynomial. Explain your answer.

- \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*}

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

- \begin{align*}3-2x\end{align*}
- \begin{align*}8x^4-x+5x^2+11x^4-10\end{align*}
- \begin{align*}8-4x+3x^3\end{align*}
- \begin{align*}-16+5f^8-7f^3\end{align*}
- \begin{align*}-5+2x-5x^2+8x^3\end{align*}
- \begin{align*}x^2-9x^4+12\end{align*}

**Mixed Review**

- Solve by graphing \begin{align*}\begin{cases} y=\frac{1}{3} x-4\\ y=-4x+10 \end{cases}\end{align*}.
- Solve for \begin{align*}u\end{align*}: \begin{align*}12=- \frac{4}{u}\end{align*}.
- Graph \begin{align*}y=|x-4|+3\end{align*}on a coordinate plane.
- State its domain and range.
- How has this graph been shifted from the parent function \begin{align*}f(x)=|x|\end{align*}?

- Two dice are rolled. The sum of the values are recorded.
- Define the sample space.
- What is the probability the sum of the dice is nine?

- Consider the equation \begin{align*}y=6500(0.8)^x\end{align*}.
- Sketch the graph of this function.
- Is this exponential growth or decay?
- What is the initial value?
- What is its domain and range?
- What is the value when \begin{align*}x=9.5\end{align*}?

- Write an equation for the line that is perpendicular to \begin{align*}y=-5\end{align*} and contains the ordered pair (6, –5).

### Answers for Explore More Problems

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