We know an awful lot about polynomials, but it relies on the very specific structure of a polynomial, and thus it is paramount that one can correctly recognize what is, and isn’t, a polynomial to use these tools.
Before we can study polynomials we need to know what they are, and before we introduce polynomials, we first need to define their individual terms, ie monomials; whose definition we give below:
Essentially a monomial is a single term with a coefficient and to non-negative a whole number (possibly zero) power. Thus terms like , and are all monomials; the last is a monomial because it can be written as .
Polynomials are just the sums and differences of different monomials. Since we will often encounter polynomials with only two terms, such as , we give those a speical name as well; binomials. Thus the expressions , , and , would all qualify as polynomials.
Keep in mind that any single term that is not a monomial can prevent an expression from being classified as a polynomial. For example, the expression is not a polynomial; even though the first two terms are both monomials, the last term () is not, and thus the overall expression is not a polynomial.
As we will see, the term with the highest power in the polynomial can provide us with a considerable information. Because of this there is a convention to write polynomials by adding the monomials starting with the largest power down to the smallest power, but this is convention only and is not always done! Be sure to double check any polynomial to see if it is written in this form or not. This convention is why we refer to the term with the highest power as the leading term, whose coefficient is the leading coefficient, and whose degree is the degree of the polynomial.
For Example: For the polynomial we could rewrite it in descending order of exponents, to get which makes clear that as the ‘leading term’ of .
For Example: For the polynomial we could rewrite it in descending order of exponents, to get which makes clear that as the ‘leading coefficient’ of .
For Example: For the polynomial we could rewrite it in descending order of exponents, to get which makes clear that as the ‘degree’ of .