Power of a Binomial

In this lesson, we delve into another important algebraic identity concerning polynomials: the power of a binomial.

In reality, the n-th power of a binomial can be calculated by manually expanding the computations. However, this technique tends to be cumbersome and prone to errors.

Instead, there exists a much quicker method to compute the power of a binomial based on the Pascal's Triangle. It's a geometric arrangement of coefficients that forms a triangle and allows for the rapid computation of binomial powers.

In this lesson, we'll explore how to construct Pascal's Triangle, its properties, and most importantly, how to apply it to calculate the power of a binomial.

Calculating the power of a binomial

In previous lessons, we studied the square of a binomial and the cube of a binomial. We can proceed sequentially and calculate the subsequent powers of a binomial. For instance, we might compute the fourth power of a binomial:

Expanding the product, we can write:

We can then compute the cube of the binomial, which we've already seen, and then multiply it by the binomial itself:

Expanding the product we get:

We've achieved the result for the fourth power, but if we wanted to calculate higher powers, we would need to repeat the procedure, which, over time, becomes quite tedious.

There exists, however, a very simple method that allows us to calculate the nth power of a binomial. This method exploits a particular geometric arrangement of numerical coefficients known as Pascal's Triangle.

Introduction to Pascal's Triangle

Pascal's Triangle is a unique geometric arrangement of numbers that finds applications in many areas of mathematics. Its distinct properties are extensively utilized in the field of combinatorial mathematics.

In this lesson, we will provide an introduction to Pascal's Triangle and focus on its application to calculating the powers of a binomial.

First, we need to see how to construct the triangle and then, how to use it to calculate the powers of a binomial.

Construction of Pascal's Triangle

Pascal's Triangle is made up of a series of rows. These rows are numbered, and by convention, the first row is row zero, i.e., . This row consists of a simple :

Subsequent rows are staggered such that the elements that compose them don't have an element directly above them but possibly two elements: one element above to their left and one element above to their right. We used the word possibly to mean that it's not given that there are always two elements above another element. To clarify, see below:

We observe three things:

1. The number of elements in a row equals the row number plus one: ;

   Indeed, row has one element, row has two, and so on.

2. We've denoted each element with two indices: . The first index indicates the row number, while the second indicates the position within the row;

   For instance, the element represents the element in the third row in the second position, while the element represents the element in the fourth row in the third position.

3. Not all elements have two elements directly above them;

   For example, the element has only one element above it, whereas the element has two elements above it. Let's say, looking at the above layout as a triangle, the elements that have only one element above them are those positioned on the sides of the triangle.

In the above layout, we stopped at row number , but in reality, Pascal's Triangle can be constructed with an arbitrary number of rows. We could have continued with row , , and so on.

Now that we know how to arrange the elements of Pascal's Triangle, we need to understand how to calculate them. The procedure is quite simple and is based on a straightforward rule:

In short, to calculate the value of an element, we need to take the elements immediately above it on the left and right and sum them. However, if one of these two elements doesn't exist, we should consider its value to be zero.

So, let's try to apply this rule and calculate the elements of the various rows step by step. Let's start with row 1:

In this case, we need to calculate the elements and . These elements have only one element above them. Therefore, consider that the value of the missing element is zero. The existing element, on the other hand, is equal to one. Thus, the elements that make up the row are both equal to . We can then write:

Let's move on to row number two:

In this case, the elements and have only one element above them. Thus, we need to consider again that the value of the missing element is zero. Their value will be equal to . On the other hand, the element has two elements above it: and . Its value will be the sum of these two elements: .

Continuing with this procedure, we obtain the following result:

Application of Pascal's Triangle

Now that we know how to construct Pascal's Triangle, we need to understand how to use it to calculate the powers of a binomial.

In fact, there is a relationship between the elements of Pascal's Triangle and the coefficients of the terms that make up the powers of a binomial. In this lesson, we will not prove this relationship. Instead, we will start with some observations. Let's examine the powers of a binomial in order.

Raising a binomial to the power of , we get:

This is an obvious result, as any quantity raised to the power of zero is equal to .

Next, raising the binomial to the power of :

This is also an obvious result, as any quantity raised to the power of is equal to itself.

Proceeding with subsequent powers:

Let's make a few observations:

1. All resulting polynomials are composed of a number of terms equal to the exponent plus one;

   Indeed, in the case of , we have only one term, in the case of , we have two terms, and so on.

2. All the resulting polynomials are almost complete polynomials with respect to the two starting monomials;

   They are almost complete in the sense that all powers of the starting monomials are present except for the constant term.

   For example, in the case of , we have , , and , but we don't have any constant term. In the case of , we have , , , and , but we don't have any constant term.

With these two observations in mind, let's arrange the results of the binomial powers so that the powers of the first monomial are arranged in descending order and the powers of the second monomial are arranged in ascending order:

As you can see, a pattern emerges. Let's highlight the coefficients:

Let's list only the coefficients:

In other words, the coefficients of the terms that make up the powers of a binomial are the elements of the corresponding row of Pascal's Triangle. Specifically, the coefficients correspond to the row number equal to the exponent.

From this, we can derive a very simple method to calculate any power of a binomial using Pascal's Triangle. Let's take the fifth power as an example:

1. First, we construct an almost complete polynomial with all the powers, from the fifth to the one with exponent 1, of the two monomials. We arrange them so that the powers of the first monomial are in descending order and the powers of the second monomial are in ascending order:

2. We take the fifth row (remembering that we number the rows starting from zero) of Pascal's Triangle:

3. We replace the coefficients of the polynomial with the elements from the row of Pascal's Triangle:

At this point, we have obtained the result.

Power of a Binomial

Let's summarise the process discussed above.

Let's look at some examples.

In Conclusion

In this lesson we have seen how to calculate the power of a binomial using Pascal's Triangle.

In particular, we've observed how to construct Pascal's Triangle and how to apply it to the computation of a binomial's power.

Starting from the next lesson, we will begin studying the division between polynomials, starting with the division of a polynomial by a monomial.