Who doesn’t like a good mystery? Challenges like “whodunnits” exercise our brains, make us think, and give us a feeling of reward when we get the right answer.

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Check it Out!Math has its own whodunnit in the form of algebraic expression. In terms of algebraic expression mysteries, you’re not looking for an answer such as, “the butler did it,” but instead trying to find a number that’s not immediately apparent.

[Picture of butler’s hands holding napkin or tray]

Solving the mystery of “what is algebraic expression?” is an exercise that will open you up to more complicated math should you choose to become one of the following:

- Engineer
- Scientist
- Doctor
- Researcher
- Project planner
- Programmer
- Coder

These jobs use complex math every day. People in these professions don’t even necessarily need a calculator to solve some of these equations, either. That’s how familiar people can get with complicated algebra.

Because many high-level jobs require knowledge of algebraic expression, we’ll give you some algebraic expression examples with answers so you can learn to solve them yourself.

## What Is Algebraic Expression?

Algebraic expressions are, at their core, part of a math problem that has some unknown numbers hidden behind letters. It helps to know some definitions.

**Variables: **These are unknown numbers represented by letters, typically x, y, or z.

**Constants: **These are the numbers we can see and can be easily recognizable numbers such as 5, 22, -4, or even 0, but could also be more complicated such as fractions like ¼, or square roots such as √8.

**Terms: **A term could be a variable on its own, a constant on its own, or a combination of the two by multiplication or division.

For example, some terms of algebraic expression are 3x or (-8÷y). When a constant is right next to a variable, such as with 3x, that implies that they will be multiplied.

**Coefficients: **These are the numbers that multiply the variables. For 3x, the coefficient would be 3.

**Like Terms: **These are terms that have the same variable, such as 5x and 10x. If your expression was 5x+10x, you could simplify this as 15x.

**Unlike Terms: **These are terms that have different variables, such as 5x and 10y. Because they are different, you can not simplify them any further. If your expression was 2x+3x+6y+4y, that could be simplified as 5x+10y, but that’s as far as you can go.

Some algebraic expression definition math examples would be 3x+9y or -22÷8x. The expression is only one part of the overall algebraic equation.

Parents, spending time learning these expressions will help your kids understand them better. Oh, and they’ll help *you* understand it better, too!

**What Is Algebraic Expression**** Simplification?**

Because algebraic expressions can get complicated when looking at all these numbers and letters, let’s take a look at some algebraic expression examples with answers. We’ll start with:

2x+3x-8y+9+14y-9y+4x

We need to look for like terms of algebraic expression and combine them. Start by combining all of the coefficients of x and y.

(2x+3x+4x)-(8y+14y-9y)+9

Then we get:

9x-3y+9

Let’s change one little thing. Instead of 8y+9, let’s make it 8y×9.

2x+3x-(8y×9)+14y-9y+4x

What is algebraic expression if not a chance for more mystery? It’s like creative writing with numbers! We’re still going to simplify for x and y and group like terms. We’ll put x at the beginning as we did before, but how we isolate y will be a little different. We can look at 8y×9 as 8(9y). It works out the same and it will make things a little easier later on.

(2x+3x+4x)-(8×9y)+(14y-9y)

Let’s add the similar elements:

9x-(8×9y)+5y

Now we can multiply the coefficients in the parentheses:

9x-72y+5y

And we get:

9x-67y

Multiplication in our algebraic expression example only made it slightly more difficult. Let’s go one step further and change one more thing. This time, we’re going to do some division:

2x+3x-(8y×9)+14y-9y+(4x÷3x)

Man, that makes it even more complex!

First, we’ll combine all our terms to isolate x and y as best we can:

5x-72y+14y-9y+(4x÷3x)

5x-67y+(4/3)

Why did those x’s disappear at the end? Because no matter what number x represents, it will ultimately end up as the fraction 4/3, or 1 and ⅓. This is an important quality to understand about algebraic expression definition math examples and will simplify these problems for you.

We can take it even further by putting in some exponents into our expression.

2x3+3x2-(8y3×9)+14x3-9y2+4x3

Well, that looks complex, right? When we asked the question, “what is algebraic expression,” you might not have thought we’d throw this much at you, but you can do it! By the way, the better you get at typing, the easier it will be to write out complicated equations like this.

First, we’ll give you a rule that will help: -(a)=-a

That rule simply shows you where you can add and subtract parentheses to make the expression easier to understand. Take away the parentheses and you get:

2x3+3x2-8y3×9+14x3-9y2+4x3

Now, let’s start simplifying by grouping like terms:

-8y3×9+2x3+14x3+4x3-9y2+3x2

Now, we can add the like terms in our algebraic expression example:

-8y3×9+20x3-9y2+3x2

And, as we’ve previously established, we can multiply 8 times 9 to get 72, simplifying our expression to:

=-72y3+20x3-9y2+3x2

There we go! We’ve simplified this expression as far as we can!

**What Are Some Different Types of ****Algebraic Expression****s?**

Now that you know the algebraic expression definition and we’ve covered some algebraic expression examples with answers, we can go over the three types of algebraic expressions.

No one expects someone just encountering these problems for the first time to get it overnight, so if it’s still a little murky, that’s ok. The more you practice creating and simplifying expressions, the easier it will become.

**Monomial Expression**

When an algebraic expression has only one term, it’s monomial. This could be 3x, 2xy, or √5x. No plus, no minus, just the single term. These expressions are already as simplified as they can be.

**Binomial Expression**

Now we’re getting a little more complicated and adding more terms of algebraic expression. Specifically, we’re up to two terms, but they must be different. An example of an algebraic expression that is *not* a binomial would be:

5x+3x

Because this can just be expressed as 8x, it doesn’t count. An example of a binomial expression would be:

5x+3y

or

5x+9

Because the expression can’t be simplified any further and only has two terms of algebraic expression, it would be a binomial.

**Polynomial Expressions**

Now we’ve arrived at three terms of expression… and beyond! A polynomial can have anywhere between three and infinity terms, but of course, we won’t be approaching infinity with our algebraic expression examples.

Since “poly” means “many,” a binomial *is* a polynomial, it’s just that because we have a definition that applies to a two-term expression, most of the time when someone says “polynomial,” they will mean three or more terms. As an example:

5x+3y+8z

An important part of polynomial expressions is that they can only contain constants, variables, and exponents. An exponent is when you want to find the square, cube, etc., and looks like this: 5xy2.

Polynomial terms can be combined using addition, subtraction, multiplication, and division. However, polynomial expressions do not allow division by a variable. So, some examples of non-polynomial algebraic expression examples would be:

5xy-3 would not be a polynomial because of the exponent -2. Exponents can only be positive numbers.

8÷y or 8/y is not allowed because this expression is divided by a variable. You could have y÷8 or y/8 because the variable divides by a constant. Take some time to actively create some polynomial expressions to simplify to flex your brain.

**Other Expressions**

Okay, so you’re not necessarily limited to three types of algebraic expression; they’re just the most common. You can also have:

**Numeric Expressions: **There’s no mystery to numeric expressions because they’re composed of numbers that can immediately be seen. For example, you might have 10+9 or 18÷6. There’s no variable to solve in numeric expressions.

**Variable Expression:** A variable algebraic expression must contain variables, numbers, and an operation. For example, you could have 5x+9 or 20y+x.

**What Is the Practical Application of ****Algebraic Expression****?**

Ah, the classic, “when will I ever need to know how to do this?” Sure, let’s make algebraic expression examples a little more concrete. After all, we said they were a mystery, so let’s create some interesting mysteries to solve! Familiarity breeds comprehension.

Remember that an algebraic expression consists of constants and variables that we add, subtract, multiply, and divide. As long as we know the rules, we can solve all kinds of real-world problems.

Let’s say we’re on a construction site, and we have a beam that is 50 feet long. We don’t know the length of two more beams, but we know that one beam is half the length of the 50 feet beam. We were also told that all three beams equal 110 feet. So we can express the problem like this:

110=50+(50/2)+x

Beam one is 50 feet; we know that. Beam two is half of 50, which we’ve expressed as 50/2. Beam three is unknown, so we’ve set that length as x. Now, let’s simplify.

110=50+25+x

Simplify further:

110=75+x

Subtract 75 from both sides, and we get:

35=x

Now we know that the length of the third beam is 35 feet. See? With a little algebra, the solution is easy to find. So the next time you ask the question, “what is algebraic expression?” Remember, there *is* an answer!