Most of the material I use for posts is information I find out about outside of school. I believe that these things should be taught in school. Lots of the posts I make are on algebra, primes, Fibonacci numbers, and other stuff that fit right into the lesson plans of teachers. It would only take two minutes or so to teach some of these things, and can make a huge impact on how much students enjoy math class.
Today, we run into a rare occasion where the post is on something I learned inside of school. It is a concept that I found cool, but my classmates didn’t. I think this was because it was presented to us as the method we have to use rather than a pattern that can make our life much easier.
In algebra, we run into situations where we need to plug a number into an equation, or find out what an equation equals if x equals a certain number. For instance, say you had the equation:
f(x) = x - 3
If we need to find out what this equation equals if x = 5, we do:
f(5) = 5 - 3
f(5) = 2
This was fairly easy. We just had a little subtraction. However, what if we had to plug 1 into this equation:
f(x) = x^5 - 3x^4 + 10x^3 + 2x^2 - 18x + 14
This is pretty complicated. You would have to go through this long process:
f(1) = (1)^5 - 3(1)^4 + 10(1)^3 + 2(1)^2 - 18(1) + 14
f(1) = 1 - 3 + 10 + 2 - 18 + 14
f(1) = 6
That wasn’t too bad, but imagine if that one were a two. Then we would have a problem. But there is a much easier way to do this.
1 | 1 -3 10 2 -18 14
_
Above, we have the number we are plugging in in that top left section. Then, we have the coefficients (the numbers before the x in the equation) lined up in order.
The first step to this method is to bring the one down below the line. Then, you multiply that one by the number in the top left. Put that directly below the -3.
1 | 1 -3 10 2 -18 14
1 _
1
Now, we must add the numbers in that second column to get the number that goes below. -3 + 1 = -2, so we put that below.
1 | 1 -3 10 2 -18 14
1 _
1 -2
And then, we do the same process as before. -2 x 1 = -2, and we put it below the 10.
1 | 1 -3 10 2 -18 14
1 -2 _
1 -2
Now, we add, and do the same thing.
1 | 1 -3 10 2 -18 14
1 -2 8 10 -8
1 -2 8 10 -8 | 6
The last number you get after adding is the answer. In this case, we got six (I sectioned it off), and as we saw, six is what we get after plugging one into the equation.
I find this interesting not because of any patterns or proofs, but because of its simplicity. There is another reason which is a little hard to explain, but it is related to division of polynomials. I hope it didn’t seem complicated, because it really isn’t. It is very cool after you think about it.
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