polynomials Flashcards
what is a polynomial?
an expression containing terms in positive integer powers of x
what is the degree of a polynomial?
the highest power of x that appears, sometimes called the order
what is a quotient and what is a remainder?
quotient is an integer multiple of a factor that goes into a polynomial, they are full units of the denominator that the polynomial has been split into
a remainder is a fragment of the numerator that doesn’t quite fulfil the denominator so is kept in fraction form
how do you divide polynomials?
do a reverse grid method.
put the denominator vertically down the left of the grid. write the highest order term on the right of the x term of the denominator in the grid. divide this by the den. x term and write the result above the grid over the larges x order. multiply the new term by the rest of the terms in the denominator.
on next row, compare what is needed to be added to each x order to get the numerator and add it to the grid diagonally upwards from its source. repeat the process
what are remainders written as?
algebraic fractions over original denominator
what is the factor theorem?
if p is a polynomial and p (a) = 0, then (x - a) is a factor of p (x)
hence to deduce if something is a factor, just sub in a to the equation and see if it = 0
how to use factor theorem with form (bx - a)?
divide a by b and sub the result in, if it = 0 then this is a factor
how do you factorise cubics?
use trial and error to find an initial factor using the factor theorem, then factorise as a normal quadratic to give 3 brackets order of 1
how do you sketch cubic graphs?
find all intersections with the coordinate axes and draw characteristic graph:
