General Form of a Polynomial
A polynomial with one variable looks like this:
|example of a polynomial
this one has 3 terms
But how do we talk about general polynomials? Ones that may have lots of terms?
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):|
So for the general case, we use this style:
And now we can say:
- an is the coefficient (the number we 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
- The Degree of the polynomial is n
- an is the coefficient of the highest term xn
- an is not equal to zero (otherwise no xn term)
- an is always a Real Number
- n can be 0, 1, 2, and so on (but not infinity)