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a train covered a certain distance at a uniform speed if the train would have been 6kmhr faster it would have taken 4hours less than the
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
the larger of two supplementary angles exceeds the smaller by 180 find them ans990810ans x y 1800 x - y
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
students are made to stand in rows if one student is extra in a row there would be 2 rows less if one student is less in a row there would be 3 rows
find the value of p and q for which the system of equations represent coincident lines 2x 3y 7 pq1x p2q2y 4pq1ans a1 2 b1 3 c1 7a2 p q
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
41x 53y 135 53x 41y 147ans 41x 53 y 135 53 x 41 y 147add the two equations solve it to get x y 3 -------1 subtract solve it to
a2bx 2a- by 2 a - 2bx 2a by 3ans 5b - 2a10ab a 10b10ab ans2ax 4ay y we get 4bx - 2by -12ax 4ay 5 4bx- 2by - 1solve this to
7y 3 - 2x 2 14 4 y - 2 3x - 3 2ans 7y 3 - 2 x 2 14 --------- 14y- 2 3x - 3 2 ----------2from
solve for x yx y - 82 x 2 y - 143 3x y - 1211 ans x2 y6ans x y - 82 x 2y - 14 3
pair of linear equations in two variables like the crest of a peacock so is mathematics at the head of all
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
if the ratios of the polynomial ax33bx23cxd are in ap prove that 2b3-3abca2d0ans let px ax3 3bx2 3cx
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
if alpha amp szlig are the zeroes of the polynomial 2x2 - 4x 5 then find the value of aalpha2 szlig2 b 1 alpha 1 szlig c alpha - szlig2 d
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
if alphabeta are the zeros of a quadratic polynomial such that alpha beta 24 alpha - beta 8 find a quadratic polynomial having alpha and beta as
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