Let c be the last non-zero digit of your monash student id number and consider the function f(x)= use mathematica to calculate the taylor polynomial of degree 5 for f(x) at x 1. use mathematica to plot f(x) for 0 gx 2, as well as the taylor polynomials of degrees 1, 2 and 3 for f(x) at x= 1 x+cx 14 marks remark. approximately one-ninth of you should be pleasantly surprised by your taylor series!

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1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.