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.

 

[blue_raver22]

First, we need to find the coefficients of the cubic polynomial p(x) in the form of ax^3 + bx^2 + cx + d. Since p(x) is a cubic polynomial, it can be written as p(x) = ax^3 + bx^2 + cx + d.

Next, we use the given information to create a system of equations. We know that p(1) = 8, so we can substitute x = 1 into our equation to get the first equation: a(1)^3 + b(1)^2 + c(1) + d = 8. Simplifying this, we get a + b + c + d = 8.

Similarly, we can substitute x = -1, 2, and -2 into our equation to get three more equations: a(-1)^3 + b(-1)^2 + c(-1) + d = -3, a(2)^3 + b(2)^2 + c(2) + d = 0, and a(-2)^3 + b(-2)^2 + c(-2) + d = 0. Simplifying these equations, we get -a + b - c + d = -3, 8a + 4b + 2c + d = 0, and -8a + 4b - 2c + d = 0.

Now, we have a system of four equations with four unknowns (a, b, c, d). We can use algebraic methods to solve for these unknowns and find the coefficients of p(x). Once we have the coefficients, we can write out the cubic polynomial p(x).

In summary, to solve this problem, we need to create a system of equations using the given information and then solve for the unknown coefficients. This may take some time and effort, but with patience and perseverance, we can find the solution.