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exponential functionas a last topic in this section we have to discuss a special exponential function actually this is so special that for
properties of f x b x1 the graph of f x will always have the point 01 or put another way f0 1 in spite of of the value of b2 for every
definition of an exponential functionif b is any number like that b 0 and b ne 1 then an exponential function is function in the
in this section we will look at exponential amp logarithm functions both of these functions are extremely important and have to be understood
find out the partial fraction decomposition of each of the following8x2 -12 x x2 2 x - 6solutionin this case the x which sits in the front is a
example find out the partial fraction decomposition of following 8x - 42 x2 3x -18solutionthe
this section doesnt actually have many to do with the rest of this chapter but since the subject required to be covered and it was a fairly short
synthetic division tablein a synthetic division table perform the multiplications in our head amp drop the middle row only writing down the third row
process for finding rational zeroes1 utilizes the rational root theorem to list all possible rational zeroes of the polynomial p x 2 evaluate the
finding zeroes of a polynomialthe below given fact will also be useful on occasion in determining the zeroes of a polynomialfactif p x is a
determine a list of all possible rational zeroeslets see how to come up along a list of possible rational zeroes for a polynomialexample find
for starters it will let us to write down a list of possible rational zeroes for a polynomial and more significantly any rational zeroes of a
example prove that the roots of the below given polynomial satisfy the rational root theoremp x 12x3 - 41x2 - 38x 40 x - 4 3x - 2 4x
if the rational number x b c is a zero of the nth degree
weve been talking regarding zeroes of polynomial and why we require them for a couple of sections now however we havent really talked regarding how
1 find out all the zeroes of the polynomial and their multiplicity utilizes the fact above to find out the x-intercept which corresponds to each
the given fact will relate all of these ideas to the multiplicity of the zerofactif x r is a zero of the polynomial p x along with multiplicity k
the humps where the graph varies direction from increasing to decreasing or decreasing to increasing is frequently called turning points if we
in this section we are going to look at a technique for getting a violent sketch of a general polynomial the only real information which were going
factor theoremfor the polynomial p x 1 if value of r is a zero of p x then x - r will be a factor of p x 2 if x - r is a factor of p x then r
if p x is a polynomial of degree n then p x will have accurately n zeroes some of which might repeatthis fact says that if you list out all the
list the multiplicities of the zeroes of each of the following polynomials p x 5x5 - 20x4 5x3
now weve got some terminology to get out of the waymultiplicity k if r is a zero of a polynomial and the exponent on the term that produced the
example determine the zeroes of following polynomialsp x 5x5 - 20x45x3 50x2 - 20x - 40 5 x 12 x - 23solutionin this the factoring has been done
well begin this section by defining just what a root or zero of a polynomial is we say that x r is a root or zero of a polynomial p x if p r