Help Robert with His Linear Algebra Problem

[Bob19]

HELP !: Linear Algebra (I'm stuck)

Hello my name is Robert and I was referred to this site by a guy from my linear algebra class.

The reason for me written to You is because I'm stuck with a linear algebra problem.

Therefore I hope You Guys can give me a hint.

I have an assignement where I am suppose to build a cubic polynomial of degree 3 using some conditions.

Here are the conditions:

p1(-2) = 2 and p'(-2) = 0 p2(4) = 6 and p2'(4) = 0

p1(0) = p2(0) and p1'(0) = p2'(0)

I know I need to construct the polynomial, but my College Professor that polynomials must be written factorized.

e.g.

p(x) = (s+t) x^3 + (s+t) x^2 + sx +t

I would very much appreciate if any of You Guys could give a hint on how to write a cubic polynomial in the for which my Professor asks.

Thanks :-)

Bob
NY

 

[NateTG]

Well, based on other, similar, questions that are flying around, I would guess, that we're looking at two cubic polynomials (rather than one which you mention in your post).

An easy way to deal with this type of exercise is to write out the cubic with variables as coefficients i.e.
$$p_1(x)=a_1x^3+b_1x^2+c_1x+d_1$$

Then write out the derivatives you have in your equations:
$$p_1'(x)=3a_1x^2+2b_1x+c_1$$

and then plug them into your equations:
$$2=p_1(-2)=a_1(-2)^3+b_1(-2)^2+c_1(-2)+d_1$$

This will produce a system of equations in the coeficients.

P.S. Since there are only 6 equations, but 8 unknowns, there will be more than one solution.
P.P.S. Cubic Polynomial and Polynomial of degree 3 mean the same thing.

 

[Bob19]

Hi Nate,

How do I then write it out in the form that my professor wants Us too ??

Best Regards,

Robert

Well, based on other, similar, questions that are flying around, I would guess, that we're looking at two cubic polynomials (rather than one which you mention in your post).

An easy way to deal with this type of exercise is to write out the cubic with variables as coefficients i.e.
$$p_1(x)=a_1x^3+b_1x^2+c_1x+d_1$$

Then write out the derivatives you have in your equations:
$$p_1'(x)=3a_1x^2+2b_1x+c_1$$

and then plug them into your equations:
$$2=p_1(-2)=a_1(-2)^3+b_1(-2)^2+c_1(-2)+d_1$$

This will produce a system of equations in the coeficients.

P.S. Since there are only 6 equations, but 8 unknowns, there will be more than one solution.
P.P.S. Cubic Polynomial and Polynomial of degree 3 mean the same thing.

 

[NateTG]

You should be able to figure out what $s$ and $t$ are if you have the polynomial's coefficient.

 

[Bob19]

I got the coefficients, but don't know how to use them to write the polynomial is desired form ??

Bob

You should be able to figure out what $s$ and $t$ are if you have the polynomial's coefficient.

 

[steven187]

hello Bob

you should have a look at the recent questions asked in this forum i think this same question was recently asked by Mathman23 have look at what was discussed in there

take care

Steven