Pascal's Triangle is defined and discussed briefly. Following the introduction to the triangle, its use in expanding polynomial powers is elaborated with examples.
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:
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!
Let's do some examples now. For the first example, see if you can use Pascal's Triangle to expand (x + 1)^7. Write out the triangle to the seventh power (remember the first line is n^0).
The lines are:
1, 2, 1
1, 3, 3, 1
1, 4, 6, 4, 1
1, 5, 10, 10, 5, 1
1, 6, 15, 20, 15, 6, 1
1, 7, 21, 35, 35, 21, 7, 1
Use the last row as our coefficients. (x + 1)^7 = x^7 + 7x^6 + 21x^5 + 35x^4 + 35x^3 + 21x^2 + 7x + 1.
Here's a tip - don't forget to reduce the exponents each time.
Let's try another example: expand (3x + 1)^3.
Don't get thrown off by the 3x here. It works the same, but now, instead of just an x, you use 3x. Like this: 1(3x)^3 + 3(3x)^2 + 3(3x) + 1 = 27x^3 + 27x^2 + 9x + 1.
Tip here - don't forget to raise the 3 of 3x to each power as well.
For our last example, another type of question you may see when working with polynomials is, 'What will be the coefficient of the fourth term in the expansion of (2x + 1)^5?'
Here, again, you can refer to Pascal's Triangle. Starting with 0 for the top, count down until you reach 5. Then you count over 4 places. You should be on the second 10. This means that for the expansion of (2x + 1)^5, the fourth term = 10(2x)^2 = 40x^2. So, the fourth term just has a coefficient of 40.
Wow, okay. We have learned that Pascal's Triangle is an amazing number pattern that identifies the coefficients for polynomial expansions.
Things that are really interesting to note about the triangle are:
- The outer numbers are all 1.
- Inside the triangle, each new number is found by adding the two above it.
- The triangle is perfectly symmetrical.
- Each line of Pascal's Triangle shows the coefficients of the expansion of the corresponding polynomial power.
This has been a very quick explanation of Pascal's triangle, focusing strictly on its use for expanding polynomials. I hope you have enjoyed the lesson and learned a lot about this great and easy-to-use tool!
Once you've completed this lesson, you should be able to:
- Describe what Pascal's Triangle is
- Identify key characteristics of Pascal's Triangle
- Explain how Pascal's Triangle is used to expand polynomials