The study of arithmetic begins (in elementary school) with the natural numbers 1, 2, 3, 4, … These are the simplest kind of number, and it is only after you learn to add, subtract multiply and divide natural numbers that you start working with other kinds of numbers like negatives, fractions, decimals, and so on.
The study of algebra, which begins when we introduce a variable like ‘x’, follows a similar path. In algebra, the most basic kind of expression is called a polynomial. We will be learning how to add, subtract, multiply and divide polynomials in the next few days. Then we will move on to more complicated kinds of expressions (rational expressions, radical expressions, and so on).
To study polynomials, we first must understand terms. A term consists of numbers and variables multiplied together, a “product of one or more numbers or variables”. The numbers can be any kind of number at all, and the number in a term is also called the coefficient of the term. The variables can be raised to powers, but only nonnegative whole number powers. A polynomial is a sum of terms, a “collection of one or more terms connected by addition or subtraction.” Here is a quick example of polynomial:
We have special names for polynomials depending on how many terms they have. A monomial has one term, a binomial has two terms, a trinomial three terms, and so on (note: the prefix ‘poly’ means ‘many’, and ‘polynomial’ is used no matter how many terms there are). Two terms are like terms if they have the same variables, raised to the same powers. The degree of a term is obtained by adding the exponents of all the variables. The degree of a polynomial is the largest degree of any of its terms.
To add polynomials, we combine like terms. When we combine terms, we add or subtract the coefficients (the numbers in front), but keep the rest of the term (variables and exponents) the same.
Subtracting polynomials is similar, but before combining like terms we must distribute the minus sign ‘-‘ through the parentheses.
Multiplying polynomials starts simple, but gets complicated as the polynomials get larger. We will start simple and work our way up to more complicated examples.
To multiply a polynomial by a monomial, you must distribute the monomial.
To multiply a binomial by a binomial, you can use the FOIL method (First, Outer, Inner, Last).
There are two special kinds of products that are common enough to have ‘shortcut formulas’ — if you remember these formulas, and can identify when to use them, it will save you some time (you can skip the FOIL). The first is the square of a binomial, and the rule is:
(a+b)^2 = a^2 + 2ab + b^2
The second rule is the sum and difference of two terms, and the rule is:
(a+b)(a-b) = a^2 – b^2
The general rule (of which all the rules above are just special cases) is this:
To multiply a polynomial by a polynomial, EACH term in the first polynomial should be multiplied by EVERY term in the second polynomial (and then add up the results).