# 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**:

Degree 2 (Quadratic) can have letters a,b,c:ax

^{2}+ bx + cDegree 3 (Cubic) can have letters a,b,c,d:ax

^{3}+ bx^{2}+ cx + d......

But for Degree "n"

**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**is the coefficient (the number we multiply by) for_{n}**x**,^{n}**a**is the coefficient for_{n-1}**x**,^{n-1}- ... etc, down to ...
**a**which is the coefficient for_{1}**x**(because x^{1}= x), and**a**which is the constant term (because x_{0}^{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**is the coefficient of the highest term_{n}**x**^{n} **a**is not equal to zero (otherwise no_{n}**x**term)^{n}**a**is always a Real Number_{n}**n**can be 0, 1, 2, and so on, but not infinity