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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
given a polynomial px along degree at least 1 amp any number r there is another polynomial qx called as the quotient with degree one less than
actually we will be seeing these sort of divisions so frequently that wed like a quicker and more efficient way of doing them luckily there is
1 determine the intercepts if there are any recall that the y-intercept is specified by0 f 0 and we determine the x-intercepts by setting the
vertical asymptotein our graph as the value of x approaches x 0 the graph begin gets extremely large on both sides of the line given by x 0 this
remember that a graph will have a y-intercept at the point 0 f 0 though in this case we have to ignore x 0 and thus this graph will never cross
find out the symmetry of find out the symmetry of
weve some rather simply tests for each of the distinct types of symmetry1 a graph will have symmetry around the x-axis if we get an equal equation
the last set of transformations which were going to be looking at in this section isnt shifts but in spite of they are called reflections amp there
we now can also combine the two shifts we only got done looking at into single problem if we know the graph of f x the graph of g x f x c
horizontal shiftsthese are quite simple as well though there is one bit where we have to be carefulgiven the graph of f x the graph of g x f x
in this section we will see how knowledge of some rather simple graphs can help us graph some more complexes graphs collectively the methods we
miscellaneous functionsthe importance of this section is to introduce you with some other functions that dont really need the work to graph that the
there are also two lines on each of the graph these lines are called asymptotes and as the graphs illustrates as we make x large in both the ve and
note that the right side has to be a 1 to be in standard form the point h k is called the center of the ellipseto graph the ellipse all that we
convert following into the convert following into the
as a last topic in this section we have to briefly talk about how to take a parabola in the general form amp convert it into the following
as noted earlier most parabolas are not given in that form so we have to take a look at how to graph a parabola which is in the general
lets go through first form of the parabola f x a x - h 2 kthere
there are two forms of the parabola which we will be looking at the first form will make graphing parabolas very simple unluckily most parabolas