12.4: Evaluate Polynomial Expressions
Have you ever thought about a cube? Take a look at this dilemma.
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.
“Hey there is a problem with these cubes,” Tanya said to her friend Michael.
Here is the problem on the sheet.
Before you is a cube. Use the formula
Tanya looked at Michael who looked at her. Both students began working on the problem in their notebooks.
You can work on this problem too. You will learn about polynomials in this Concept. By the end of the it, you will be able to solve this problem.
Guidance
A polynomial is an algebraic expression that shows the sum of monomials.
Since the prefix mono means one, a monomial is a single piece or term. The prefix poly means many. So the word polynomial refers to one or more than one term in an expression. The relationship between these terms may be sums or difference.
Polynomials:
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.”
You have learned the order of operations, commonly called PEMDAS. In other words, arithmetic operations are performed in the following order:
 First any operations inside grouping symbols (P).
 Second any values with exponents (E).
 Third multiplication and division in order from left to right (M and D).
 Finally addition and subtraction in order from left to right (A and S).
When we consider expressions, we can evaluate an expression for a given value. In other words, we can find the total value if we know how much the variable is. We can replace the variable(s) with the given value and then use the order of operations to calculate the total value.
Take a look at this one.
Evaluate
Step 1: Replace the variables with the given value, 5.
Step 2: Find the total value using the order of operations.
This is our answer. We can evaluate any expression when we have been given a value for the variable.
Evaluate each expression by using the given value.
Example A
Evaluate
Solution:
Example B
Evaluate
Solution:
Example C
Evaluate
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
Now you can use the formula and the given information to solve for the surface area of the cube. The given length of the cube is 8 feet. You can substitute this into the formula for the side length.
This is the surface area of the cube.
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
 Like Terms
 are terms that have the same variables and same exponents.
Guided Practice
Here is one for you to try on your own.
Evaluate
Solution
First, substitute the given value into the expression for
Next, simplify according to the order of operations.
Our answer is
Video Review
Practice
Directions: 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 
12x2+8x+11 forx=2 
6y2−2y−8 fory=6 
\begin{align*}3(x7) + 5(x + 1)\end{align*}
3(x−7)+5(x+1) for \begin{align*}x = 10\end{align*}x=10 
\begin{align*}2y^3+6(y4)+y\end{align*}
−2y3+6(y−4)+y for \begin{align*}y = 3\end{align*}
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.like terms
Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.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.Trinomial
A trinomial is a mathematical expression with three terms.Image Attributions
Here you'll identify and evaluate polynomial expressions.