1. ## Polynomials

Find all the polynomials f(t) of degree less than or equal to 3, such that f(0) = 3, f(1) = 2, f(3) = 0 and the integral from 0 to 2 of f ' (t) dt = 4

- Help with this problem would be appreciated a lot. Thanks.

2. You have to solve a 4x4 system of linear equations.

The polynomial is of the form $ax^{3}+bx^{2}+cx+d.$ The first three data give you three of the equations you need in order to determine a, b, c, and d.

Since the integral of the derivative of f between 0 and 2 is equal to f(2)-f(0), you have a fourth equation here. So, have a happy Gauss-elimination (or whatever way you call it) practice!

3. I dont get it..

So my matrix will look something like this?

a b c 3
a b c 2
a b c 0

How do I determine what a, b and c are?

4. No you plug in your values into your 3rd degree polynomial:

$
ax^{3}+bx^{2}+cx+d.
$

so f(0)=3:

$a(0)^{3}+b(0)^{2}+c(0)+d=3$

f(1)=2

$a(1)^{3}+b(1)^{2}+c(1)+d=2$

f(3)=0

$a(3)^{3}+b(3)^{2}+c(3)+d=0$

f(2)-f(0)

$f(2)-f(0) = a(2)^{3}+b(2)^{2}+c(2)+d-(a(0)^{3}+b(0)^{2}+c(0)+d)=8a+4b+2c = 4$

$\left [\begin{array}{cccc}
0 & 0 & 0 & 1 \\
1 & 1 & 1 & 1 \\
8 & 4 & 2 & 0 \\
27 & 9 & 3 & 1 \end{array} \right]*\left [\begin{array}{c}
a \\
b \\
c \\
d \end{array} \right]=\left [\begin{array}{c}
3 \\
2 \\
4 \\
0 \end{array} \right]$

I put the f(3) equation in the last row btw.

5. OH, wow. Now this looks very clear. The matrix is not the hard part, its setting it up I dident understand. My first course in Linear. Thanks to both of you!!

6. no problem it made me learn how to type out a matrix in Latex . . . though if you want me to show the elimination if you run into trouble with that I don't know how long that'll take me to type.

7. haha, no thats just basics. The diagnal row has to be 1's and everything zeros from what i understand.

8. it's basic but its' also easy to mess up