Mr. Travis is taking his Social Studies class on a tour of downtown. He has created a scavenger hunt for the students as they travel around the city. The scavenger hunt is made up of all different types of architecture and landmarks as well as problems that will need to be solved. Mr. Travis asked the bus to drop the students off in front of the Town Hall. In the square across from the town hall is a plaza with three cubes in it.

One of the problems on the sheet read “Before you is a cube. Use the formula

Can you solve this problem?

In this concept, you will learn to evaluate polynomial expressions.

### Evaluating Polynomials

A **polynomial** is an algebraic expression that shows the sum of monomials.

In order to solve algebraic expressions, you need to use the order of operations, commonly called PEMDAS. The diagram below shows the order of arithmetic operations performed using PEMDAS.

When you consider expressions, you can evaluate an expression for a given value. In other words, you can find the total value if you know the value that corresponds to the variable. You can replace the variable(s) with the given value and then use the order of operations to calculate the total value.

Let’s look at an example.

Evaluate

First, replace the variables with the given value of

Next, find the total value using the order of operations.

The answer is 30.

### Examples

#### Example 1

Earlier, you were given a problem about the cube. You have been asked to find the surface area of a cube that has a side length of 8 feet.

First, substitute the side length into the formula for the surface area.

Next, use PEMDAS to solve for the surface area.

The answer is 384.

The surface area of the cube is 384 ft^{2}.

#### Example 2

Evaluate

First, replace the variables with the given value of

Next, find the total value using the order of operations.

The answer is 57.

#### Example 3

Evaluate

First, replace the variables with the given value of

Next, find the total value using the order of operations.

The answer is 23.

#### Example 4

Evaluate

First, replace the variables with the given value of

Next, find the total value using the order of operations.

The answer is 3.

#### Example 5

Evaluate

First, replace the variables with the given value of

Next, find the total value using the order of operations.

The answer is 29.

### Review

Evaluate the following expressions for the given value.

7x3 forx=2 6x2 forx=3 4x3 forx=2 8x2 forx=2 10xy forx=2,y=3 7x2+4x forx=2 6x2+5x forx=2 3x2+8x forx=3 7x2+4x−2 forx=2 9x2+5x−3 forx=3 5x2+5x−2 forx=2 - \begin{align*}12 x^2 + 8x + 11\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}6y^2 - 2y -8\end{align*} for \begin{align*}y -6\end{align*}
- \begin{align*}3 (x-7) + 5 (x+1)\end{align*} for \begin{align*}x=10\end{align*}
- \begin{align*}-2y^3 + 6 (y-4) + y\end{align*} for \begin{align*}y =-3\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.4.