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Remedial Algebra I25 chapters | 248 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In algebra, you come across various kinds of functions. You start learning about linear functions, and then quadratic functions. Now, though, you come across cubic functions. What are they? Watch this video lesson to find out.

As you progress in your studies of algebra, you are exposed to more and more types of equations. You now are familiar with linear and quadratic equations. You know what they are and you can identify them easily. You can even tell a friend why they are important to know.

Now that you know both linear and quadratic equations like the back of your hand, it is now time to learn about cubic equations. What are they? Why are they important in life? Keep watching this video lesson, and we will find out!

First off, **cubic equations** are equations with a degree of 3. This means that the highest exponent is always 3. In algebra, we can write their general form as *ax*^3 + *bx*^2 + *cx* + *d* = 0, where *a*, *b*, *c*, and *d* are numbers, with the one restriction that *a* cannot be 0. So, cubic equations can have just one term as long as it has an exponent of 3.

It can also have up to four terms. For example, 4*x*^3 = 0 is a cubic equation, as is 4*x*^3 + 3*x*^2 + 2*x* + 1 = 0. Notice that both of these cubic equations have that little 3 as the highest exponent. It is this little 3 that you will always look for when you want to identify a cubic equation.

The equation *x* + 2*x*^3 + *x*^2 = 0 is also a cubic. Notice the little 3 that is our highest exponent. This equation just happens to be written out of order. We can rewrite it in the standard form by writing the *x*^3 term first, followed by the *x*^2, and so on. Rewriting it, we get 2*x*^3 + *x*^2 + *x* = 0.

Cubic equations come in all sorts. All of these are examples of cubic equations:

*x*^3 = 0- 2
*x*^3 + 4*x*+ 1 = 0 - 4
*x*^3 +*x*^2 + 4*x*- 8 = 0

Do you see that all of these have the little 3? Just remember that for cubic equations, that little 3 is the defining aspect. Now, let's talk about why cubic equations are important.

In algebra, cubic equations have been around for centuries. The ancient Babylonians had ways of calculating cubic equations. Archeologists have found really old Babylonian tablets that showed tables that helped people back then to solve cubic equations. This goes all the way back to the 20th century BC. Now that's really old! Since then, mathematicians over time have added to the knowledge of solving cubic equations. We now have several methods to solve them thanks to all the hard work of mathematicians over the years.

While cubic equations have been a big part of algebra, they've also played an important part in our real-world problem solving. One of the biggest applications that we have of cubic equations is one that you are most likely very familiar with and one that you probably didn't even think to link with cubic equations. What is this application?

It is that of finding the volumes of various objects. Think about the formula for finding the volume of a cube. What is it? Is it a cubic equation? It definitely is.

The equation for the volume of a cube, where all the sides are the same measurement, is *V* = *s*^3, where *s* is for the length of a side. Do you see the little 3 as the highest exponent?

Another example is that of a sphere, *V* = (4 * pi * *r*^3) / 3. Do you see the little 3 here as well? This is the biggest application of cubic equations in real life. There are more applications that you can research on your own later.

For now, we have reached the end of our video lesson. But before we finish, let's review what we have learned. We've learned that a **cubic equation** is an equation of degree 3.

To identify them, we look for the little 3 as our highest exponent. Cubic equations can have just one term or they can have up to four. Their general form is *ax*^3 + *bx*^2 + *cx* + *d* = 0, where *a*, *b*, *c*, and *d* are numbers, and *a* cannot be 0.

Cubic equations have been an important part of algebra for hundreds of years dating back to the time of the ancient Babylonians in the 20th century BC. Since then, mathematicians have made advancements in the study of cubic equations to the point where we now have different ways of solving them. In the real world, one major application of cubic equations is that of finding the volumes of various objects.

The information presented in this video lesson could prepare you to:

- Identify cubic equations
- Understand their importance
- Give examples of their real-life applications

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Remedial Algebra I25 chapters | 248 lessons | 1 flashcard set

- What is a Cubic Equation? - Definition & Examples 5:23
- How to Solve Equations that are Not Perfectly Cubed 7:34
- Using the Greatest Common Factor to Solve Cubic Equations 4:01
- Grouping to Factor Cubic Equations 6:41
- Solving Cubic Equations with Integers 8:33
- Changing Radical Equations into Linear or Quadratic Equations 5:14
- Squaring Both Sides of an Equation Twice 3:42
- Go to High School Algebra: Cubic Equations

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