# 12.4: Evaluate Polynomial Expressions

**Basic**Created by: CK-12

**Practice**Polynomial Expression Evaluation

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 \begin{align*}A = 6s^2\end{align*} to find the surface area of a cube whose side measures 8 feet.

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**: \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.”**

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 \begin{align*}x^2+3x-10\end{align*} for \begin{align*}x=5\end{align*}

**Step 1:** Replace the variables with the given value, 5.

\begin{align*}5^2+3 \cdot 5-10\end{align*}

**Step 2:** Find the total value using the order of operations.

\begin{align*} & 5^2+3 \cdot 5-10\\ & 25+3 \cdot 5-10 && (\text{There is no group, so first is the number with the exponent.})\\ & 25+15-10 && (\text{Complete the multiplication})\\ & 40-10 && (\text{Addition and subtraction from left to right.})\\ & 30 && (\text{Our total is} \ 30.)\end{align*}

**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 \begin{align*}x^2+5x-1\end{align*} for \begin{align*}x=3\end{align*}

**Solution: \begin{align*}13\end{align*}**

#### Example B

Evaluate \begin{align*}x^2+4x-9\end{align*} for \begin{align*}x=2\end{align*}

**Solution: \begin{align*}3\end{align*}**

#### Example C

Evaluate \begin{align*}2x^2+2x+5\end{align*} for \begin{align*}x=3\end{align*}

**Solution: \begin{align*}29\end{align*}**

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.**

\begin{align*}A &= 6s^2\\ A &=6(8^2)\\ A &=6(64)\\ A &=384 \ sq.feet \end{align*}

**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 \begin{align*}4x^2+2x+15\end{align*} for \begin{align*}x=3\end{align*}

**Solution**

First, substitute the given value into the expression for \begin{align*}x\end{align*}.

\begin{align*}4(3)^2+2(3)+15\end{align*}

Next, simplify according to the order of operations.

\begin{align*}4(9)+2(3)+15\end{align*}

\begin{align*}36+6+15\end{align*}

\begin{align*}57\end{align*}

**Our answer is \begin{align*}57\end{align*}.**

### Video Review

### Practice

Directions: Evaluate the following expressions for the given value.

- \begin{align*}7x^3\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}6x^2\end{align*} for \begin{align*}x=3\end{align*}
- \begin{align*}4x^3\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}8x^2\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}10xy\end{align*} for \begin{align*}x=2,y=3\end{align*}
- \begin{align*}7x^2+4x\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}6x^2+5x\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}3x^2+8x\end{align*} for \begin{align*}x=3\end{align*}
- \begin{align*}7x^2+4x-2\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}9x^2+5x-3\end{align*} for \begin{align*}x=3\end{align*}
- \begin{align*}5x^2+5x-2\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}12x^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*}

### Notes/Highlights Having trouble? Report an issue.

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Term | Definition |
---|---|

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.