Polynomials are the next step after lines and quadratics. It refers to sums of terms of the form of a number multiplying a positive integer power of an unknown. Examples include:
For polynomials, we generally name them using letters like {$$}p{/$$}, {$$}q{/$$}, or {$$}r{/$$}. The degree of a polynomial refers to the highest power of {$$}x{/$$} (or whatever variable name is of relevance).
The expression {$$}p(x){/$$} is read as "p of x" and is meant to communicate that we are to put in a value in for {$$}x{/$$}. To evaluate the polynomial {$$}p{/$$} at 3, for example, we would have
{$$} p(3) = 53^3 -73^2 +2*3 + 1 = 79 {/$$}
Polynomials can refer to single terms or many terms though the topic generally refers to those with at least degree 3 since degree 1 (lines) and degree 2 (quadratics) are covered separately.
This chapter covers the basic combination of polynomials, the questions we may care about regarding polynomials such as finding values that make a polynomial zero, and exploring deeply a fantastic tool called synthetic division. A novel approach here is to explore some notions of calculus using synthetic division.
The numbers in front of the power of {$$}x{/$$} is called a coefficient. Coefficients can be any number we like, however, the coefficient of the leading degree term should not be zero. This is similar to how we do not write 0300 for the number three hundred.
The coefficient and the power of {$$}x{/$$} combine to be a monomial. It is often also referred to a term, but a term can also refer to the portion of the monomial with no coefficient. This can be useful as we can talk about the number of terms in a polynomial in a way which is not subject to different ways of breaking apart the monomials.
Adding, subtracting, multiplying polynomials