multiplicity qn help (1 Viewer)

[Masaken]

I've differentiated P'(x) and then P''(x), then did P''(2). I'm not sure if this is 100% correct so far or what to do next, what would you do?

 

[pseo0123]

You substitute x=2 into values p(x), p'(x) and p''(x) and since there's a triple root at this value, these 3 equations are all equal to 0. You should get the following equations:
1. p(2): 4a+2b+c=0
2. p'(2): 32+12a+4b+c=0
3. p''(2): 24+6a +b = 0
(the equations are all simplified)

for equation 3, we only have 2 values so we can find one in terms of the other. I chose to find b in terms of a which is:
b=-24-6a

Then, you substitute this into the 2nd equation because we want to simplify the 2nd equation into another 2 values. You get the next equation:
c=64+12a
when simplified

Now we have 2 new equations: b in terms of a and c in terms of a. There's one last equation we haven't used which is the first one and all we have to do is substitute these 2 new equations into the values for b and c. Once you simplify you get that a = -4. After that, you can just solve for the other equations so b=0, c=16 and that's it

 

[Masaken]

Thanks!