3.3 Polynomial and Synthetic Division

Long Division: Let’s Recall

Same method if it’s a polynomial! *** The polynomial must be in standard form; if a term is missing, it must be written in with a coefficient of 0! ***

The Division Algorithm If f(x) and d(x) are polynomials such that and the degree of d(x) is less than or equal to the degree or f(x), there exists unique polynomials q(x) and r(x) such that where r(x) = 0 or the degree of r(x) is less than the degree of d(x). If the remainder r(x) is zero, d(x) divides evenly into f(x)

It can also be written like this:

Synthetic Division: can only be used for divisors of the form x-k (or x - (-k)) Divide

Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is

Factor Theorem: A polynomial f(x) has a factor (x-k) if and only if Show that (x-2) and (x+3) are factors of

Using the Remainder in Synthetic Division The remainder r, obtained in the synthetic division of f(x) by x-k, provides the following information: 1-The remainder r gives the value of f at x = k. That is,. 2-If r = 0. (x - k) is a factor of f(x). 3-If r = 0, (k, 0) is an x- intercept of the graph of f.

What if I just asked you to factor from without giving you anything else? How do you know what to put on the outside?

The Rational Zero Test If the polynomial has integer coefficients, every rational zero has the form Rational zero = Where p and q have no common factors other than 1, and p = a factor of the constant term q = a factor of the leading coefficient