General Form of a Polynomial

A polynomial with one variable looks like this:

polynomial example
example of a polynomial
this one has 3 terms

But how de we talk about general polynomials? Ones that may have lots of terms?

General Form

A general polynomial (of one variable) could have any number of terms ...

... for Degree 2 (Quadratic) we can use the letters a,b,c:   ax2 + bx + c
     
... also Degree 3 (Cubic) can have letters:   ax3 + bx2 + cx + d
     
...   ...
     
... but for Degree "n", using letters won't work:   axn + bxn-1 + ... + ?x + ?
 
The trouble is, we don't know what letters to end on!

 

So instead of "a, b, c, ..." we use the letter "a" with a little number next to it (which says which term it belongs to): polynomial general term

So for the general case, we use this style:

polynomial general form

And now we can say:

  • an is the coefficient (the number you multiply by) for xn,
  • an-1 is the coefficient for xn-1, etc,
  • ... down to ...
  • a1 which is the coefficient for x (because x1 = x), and
  • a0 which is the constant term (because x0 = 1).

Example: 9x4 + 5x2 - x + 7

  • a4 = 9
  • a3 = 0 (there is no x3 term)
  • a2 = 5
  • a1 = -1
  • a0 = 7

Note also:

  • The Degree of the polynomial is n
  • an is not equal to zero (because then there would be no xn term)
  • an is always a Real Number
  • n can be 0, 1, 2, and so on (but not infinity)