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Now that we know what a binomial expression is, let's multiply a binomial expression with itself to get powers of the binomial term. These powers can be expanded using the binomial theorem.
A general expression for binomial theorem goes as follows:
A power of
Let’s look at a simple binomial expression with the first few powers:
In the following section, we'll give a visual explanation of some simple binomial expressions, without going into the algebraic manipulations.
We add the two terms
The above expression involves squares and multiplications. In geometry, the multiplication of two terms defines the area of a shape. The most commonly used areas are the areas of squares and rectangles.
The area of a square shape is simply the square of the length of its side.
The area of a rectangle is the multiplication of its length with its width.
In the equation above,
We see the square terms,
The term involving
Let’s visualize it in the following figure:
We can see that the areas of the two regions in the image above are the same.
The side of the square with the length
By adding all these areas we get the area of the larger square that is
Equating the areas of the larger square to the sum of areas of the components, we get the binomial expansion of power
Let’s make a slight variation in the binomial terms. Instead of taking the sum, we take the difference
Let’s show it geometrically.
Visualizing it geometrically is not as simple as in the case of the sum of the binomial terms. Again, we see squares and multiplication terms, indicating we can visualize with squares and rectangles and compute their areas.
On the left-hand side of the equation, we see the difference of the two terms. We can think of it as the difference between the lengths of the sides of two squares. The larger square has the side of length
Let’s look at the right side of the figure now. We get a larger square with a side of length
It contains the smaller square of side
On top of this smaller square is a rectangle with sides
The same rectangle is there, in the vertical position on the right side of the square with sides
We subtract areas of the rectangles from the larger square of the side
It’s important to notice here that the two rectangles overlap with each other. The area of the intersection is
So the two areas are the same:
Let’s look at another binomial expression:
This gives us a nice factored form when we have a difference of two squared terms. We can easily show it algebraically by simply multiplying the factors on the right-hand side of the equation. Let’s show it visually.
The left-hand side of the equation indicates the difference of areas of two squares
On the left side of the figure, we see the difference in the areas of the two squares. We get a figure whose upper-right corner is missing. Let’s see how an area of this shape equals the area of a rectangle with sides
On the right side, we see the blue rectangle has sides of lengths
Below the square with area
The two rectangles and the square, when stacked on top of each other, form a larger rectangle with sides of lengths
So, we get the identity:
Can we do it for higher powers? Let’s see for
We can validate it simply by multiplying the
Let’s visualize it geometrically.
We can think of
We get a number of cubes and cuboids. Let’s gather their volumes.
We see two cubes of volumes
There are three cuboids of volumes
There are three cuboids of volumes
We get the total volume as follows:
Note: For higher powers it becomes hard to show it visually on a two dimensional plane.
Geometry gives a beautiful way to understand the basic mathematical concepts. In this blog, we proved some basic binomial theorem expansions with the help of geometric shapes. Although it can be done algebraically, the geometric proofs give us a good insight into how the mathematical expressions are actually working.
Getting insights with geometric proofs may help to visualize many mathematical problems and analyses of algorithms.
Problem solving skill is fundamental to computing. Visualizing a problem and its solution simplifies the problem solving approach, its explanation, and analysis. Educative offers some interesting courses to develop and enhance problem solving skills. Have a look at the following courses offered here at Educative:
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