A polynomial is an expression which involves various
powers of a variable. Some examples are:
Here the coefficients (a's) can be any real number. The number n, which is an integer, is known as the degree of the polynomial.
EX. is a polynomial of
degree 3, with leading coefficient 5, and constant term -2.
Polynomials can be classified based on the degree (largest power), or based on the number of terms.
1 term = monomial
EX. is a binomial of degree 5.
Basic Operations on Polynomials
Just like with numbers, we often want to add, subtract, multiply and divide polynomials.
Adding polynomials is simply collecting like terms
To subtract two polynomials, you simply use the distributive law, then collect like terms.
Notice you distribute the minus sign.
While adding and subtracting were fairly easy, multiply two polynomials can quickly become a headache. To multiply you must repeatedly apply the distributive law, then collect like terms.
Notice that the distributive law was applied four times to complete that single multiplication
There is a special case for multiplying polynomials, which occurs when you multiply a binomial times a binomial. In this case it is easier to use FOIL.