# Polynomials

A polynomial (in one variable) is something of the form

The a's stand for numbers, and x is the variable (but you can use any letter for the variable). In other words, a polynomial is a sum of nonnegative powers of the variable, possibly multiplied by numbers. (The term is , so it fits this description.)

Example. Here are some polynomials:

Notice that 17 --- a number by itself --- is considered a polynomial.

The following are not polynomials:

You can also have polynomials in several variables. Just as with polynomials in one variable, you can have a sum of terms, each consisting of a product of positive integer powers of the variables and numbers.

Example. Here are some polynomials:

The following are not polynomials:

For the most part, I'll concentrate on polynomials in one variable.

You add and subtract polynomials by grouping like terms, then adding or subtracting.

Example.

If is a polynomial in the variable x, the value of at is found by plugging c in for x in . The value of at is denoted .

Example. Find the value of when and when .

If , then c is a root of . In this case, 2 is a root of .

You can multiply polynomials vertically, the way you multiply numbers.

Example.

Thus,

Example.

Thus,

Example.

Hence,

Example.

Therefore,

The last example is a special case of a factoring formula I'll discuss later.

Example. You can also multiply polynomials involving more than one variable.

Thus,

You can also multiply polynomials horizontally. Consider the case . The FOIL method produces the product term-by-term:

Thus,

Example.

Example.

FOIL also works with polynomials of two or more variables.

Example.

Some special forms come up often enough that it's worth listing them.

1.

2.

3. .

Example.

Example.

Example.

Example.

Example.

Example.

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