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?
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: | ax^{2} + bx + c | |
... also Degree 3 (Cubic) can have letters: | ax^{3} + bx^{2} + cx + d | |
... | ... | |
... but for Degree "n", using letters won't work: | ax^{n} + bx^{n-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:
- a_{n} is the coefficient (the number we multiply by) for x^{n},
- a_{n-1} is the coefficient for x^{n-1}, etc,
- ... down to ...
- a_{1} which is the coefficient for x (because x^{1} = x), and
- a_{0} which is the constant term (because x^{0} = 1).
Example: 9x^{4} + 5x^{2} - x + 7
- a_{4} = 9
- a_{3} = 0 (there is no x^{3} term)
- a_{2} = 5
- a_{1} = -1
- a_{0} = 7
Note also:
- The Degree of the polynomial is n
- a_{n} is the coefficient of the highest term x^{n}
- a_{n} is not equal to zero (otherwise no x^{n} term)
- a_{n} is always a Real Number
- n can be 0, 1, 2, and so on (but not infinity)