Polynomials

There are many different types of algebraic expressions, but most of the expressions in this class are a particular type, called a polynomial. A polynomial is any expression built out of constants and variables using the operations of addition and multiplication, or anything equivalent to this. (You don't have to use all of these in any given polynomial.) Each polynomial has a standard form; usually the first step in dealing with any expression is to put it in standard form.

Standard forms

Every polynomial is a sum of terms; you add up the terms to get the entire expression. Here are a couple of examples: The last example shows that, when analysing polynomials, you want to think of subtraction as a form of addition. Here, we think of subtracting 2x2 as adding −2x2.

Each term of a polynomial is a monomial; a monomial is any expression built out of constants and variables using only the operation of multiplication, or anything equivalent to this.

Every monomial is a product of factors; you multiply the factors to get the entire term. The first factor is the coefficient, which is always a constant (no variables). Then there is one factor for each variable, which consists of that variable raised to the power of a whole number. This whole number is the degree of the monomial on that variable. So in the end, a monomial is given by several numbers: the coefficient and the degrees on all of the variables.

Think about these special cases before we look at some examples:

Now here are some examples:

Now if you look back at the first two examples of polynomials, we see that each of these polynomials has three terms, and each of these terms has a coefficient and a degree on x.

So far, I've only talked about degrees on specific variables. The degree of a monomial is the sum of the degrees for the variables in the monomial. For example:

When writing out a polynomial, we always write the terms in order of decreasing degree. The degree of the polynomial as a whole is the largest degree of any of its terms. Returning to our original examples:

Operations on polynomials

Two terms are alike (or like terms) if they are the same except for their coefficients. For example: You can combine like terms into a single term by adding the coefficients; for example: This also defines addition of polynomials; for example:

Similarly, two factors are alike (or like factors) if they are both coefficients or involve the same variable. For example:

You can combine like factors into a single factor by adding the degrees; for example: This also defines multiplication of monomials; for example: Finally, to define multiplication of polynomials, we multiply each term by each term. For example: This example also shows the importance of putting the terms of a polynomial in order of decreasing degree.
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