Copyright

Pascal's Triangle: Definition and Use with Polynomials

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: The Binomial Theorem: Defining Expressions

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:03 Pascal's Triangle
  • 1:51 Polynomial Uses
  • 4:06 Examples
  • 6:32 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Pascal's Triangle is defined and discussed briefly. Following the introduction to the triangle, its use in expanding polynomial powers is elaborated with examples.

Pascal's Triangle

Have you ever been to Egypt? I haven't, but it is definitely one of the places in this world that I would like to visit. Thinking of Egypt always brings to mind pyramids! Pyramids evoke a sense of mystery and amazement in me.

In this lesson, we will be learning about the mysterious and amazing Pascal's Triangle. Pascal's Triangle is an amazing number pattern that creates a pyramid, or triangle, shape out of the binomial coefficients. It is named after the French mathematician.

The triangle starts with a 1 at the top, the pinnacle of the pyramid. The next line in the triangle has two 1s: like this:

null

The third line of the triangle is 1, 2, 1. This is where it starts to get really interesting. Let's put in two more lines of the triangle to really see what is going on.

Did you notice that the outside edges are always 1? Did you notice that inside the triangle, each new number is found by adding the two above it?

If you didn't notice the adding aspect, look again. On the second line we have 1 and 1. Just under those two ones is a 2. 1 + 1 = 2. On the next line down we have a 1 and a 2 next to each other. Under them is a 3. Under the two 3s is a 6. Do you see it now? This pattern continues for the whole triangle. Using this adding process, you could create Pascal's Triangle for yourself out to however many lines you'd like.

The last thing to notice, and it will be even more obvious as you build the triangle, is that the triangle is perfectly symmetrical. So, as you are building the lines, remember, once you reach the middle, the rest of the numbers for any line will match the reverse of the first half of the line.

Polynomial Uses For Pascal's Triangle

Okay, so, Pascal's Triangle is a pretty cool pattern of numbers, but how does that help with polynomials?

Before we answer that, let's look at a polynomial expansion: (x + 1)^3. In order to work out this problem we have to multiply: (x + 1)(x + 1)(x + 1). Eventually, we would end up with: x^3 + 3x^2 + 3x + 1.

If you aren't sure how I got that answer, please review some of the other algebra lessons.

What I notice about this result is:

  • Starting with the original power, the powers decrease by one in each term
  • The coefficients for each term are 1, 3, 3, 1

Wait, 1, 3, 3, 1! That is one of the lines of the triangle!

Are you ready for this? Each line in the triangle actually represents the coefficients of a polynomial expansion!

Looking back at our triangle, notice how the polynomial expansion coefficients match up perfectly with the triangle numbers:

  • (x + 1)^0 = 1
  • (x + 1)^1 = 1x + 1
  • (x + 1)^2 = 1x^2 + 2x + 1
  • (x + 1)^3 = 1x^3 + 3x^2 + 3x + 1
  • (x + 1)^4 = 1x^4 + 4x^3 + 6x^2 + 4x + 1

This illustration makes the pattern pretty easy to see. If you wanted to expand (x + 1)^5, you would just need to add another line to the triangle: 1, 5, 10, 10, 5, 1. Thus the expansion is x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.

All you have to remember is to decrease the exponents for each term, and the rest is done for you in the triangle!

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create An Account
Support