HELP Detrivative matrix/system of equations

[theacerf1]

Iv been stuck on this question for hrs now, no idea how to solve it. Can someone please explain, thanks! :)

Let p(x) be a cubic polynomial and p′(x) be its derivative, and suppose that p(1) = 8,
p(−1) = −3, p′(2) = 0 and p′(−2) = 0. Find p(x).

 

[spamiam]

Hello and welcome to the forum! First of all, if this is a homework question, you should post it in the homework section of the forum.

If this isn't a homework question, I'll be happy to post my solution, but I'll wait for your response. You could start by writing p(x) = ax³ + bx² + cx + d and trying to solve for the unknowns using your restraints.

 

[theacerf1]

Hey spamiam, no its not a homework question, iv got my exam nxt week for math and just working through different textbooks solving different problems. The textbook only gives the answer but what I want to know is how they got the answer, basically the working out as i have no idea how to solve this kind of question if it pops up in the exam :(

Yeah, wat u wrote is all i could basically think of too...
p(x) = ax3 + bx2 + cx + d
p'(x) = 3ax^2 + 2bx + c

then subbing p(1) = 8, p(−1) = −3, p′(2) = 0 and p′(−2) = 0 to get 4 equations. Then I am totally stuck. There is more questions just like this so if i can work out exactly how to do this, i can attempt the other ones with different numbers.

Thanks for the help and quick reply :)

 

[spamiam]

Well that's a good start: 4 equations with 4 unknowns. Here's what I got

(1) a + b + c + d = p(1) = 8

(2) -a + b - c + d = p(-1) = -3

(3) 12a + 4b + c = p'(2) = 0

(4) 12a - 4b + c = p'(-2) = 0

You can just use Gaussian elimination, but I think it might be easier to just solve this particular system using a couple tricks. Subtracting (4) from (3), I get 8b = 0, so b=0, simplifying things significantly. Adding (1) and (2) together, I get 2b + 2d = 5, so b + d = 5/2. Since b=0, then d = 5/2. Using (3) with b=0, we have c = -12a. Subtracting (2) from (1), we have 2a + 2c = 11, so a + c = 11/2. Substituting in c = -12a, we get a = -1/2, so c = 6. so p(x) = -\frac{1}{2}x^3 +6x + \frac{5}{2}.

Anyway, there are no real fixed rules for solving systems of equations, aside from Gaussian elimination, so you just have to play around with them to eliminate variables.