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Polynomials Explained in 5 Minutes

Polynomials Explained in 5 Minutes

One of the simplest things you can do with a number is multiply it by itself. If x is the number, we have..


Mathematicians use a shorthand for this. They write the number x with a small superscript 2 in the upper right corner like this..


They describe this in several ways. Sometimes they say the number has been "squared", sometimes they say the number has been "raised to the power of 2".

This notation immediately suggests x^3=x*x*x and x^4=x*x*x*x and so on. So we can raise a number to any power, or at least any positive integer power. For example, x raised to the power of 7 means this: take 7 x's and multiply them all together x^7=x*x*x*x*x*x*x and of course x^1=x

Notation in mathematics is important, it can help manipulate objects, it can save time, and it can help introduce new ideas.

For example, with the above definition we can raise any number to a power that's a positive integer {1,2,3,4...} but can we extend this?

A clue comes from this simple fact..

(x^a)*(x^b)= x^(a+b)

Where a and b are positive integers. So what about this...


x^0=1 for any x, and this is easy to prove..

(x^0)*(x^1)=x^(1+0)=x^1=x which can only be true if x^0=1

What about negative powers, for example, what's x^(-1)

Well, x^(-1)*x^1=x^(-1+1)=x^0=1

So x^(-1)=1/(x^1)=1/x

Powers also allow us to write a whole new set of equations. For example..

x^2-9=0 which has the solutions x=3 and x=-3
x^5=32 which has the solution x=2
x^4=1 which has the solutions x=1 and x=-1
x^4+x^3+x^2+x-4=0 which has the solution x=1

Equations like this, made from powers, are called polynomial equations.

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Content written and posted by Ken Abbott