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Verify that the polynomials

$p(x) = 5x^3 - 27x^2 + 45x - 21$

$q(x) = x^4 - 5x^3 +8x^2 - 5x + 3$

Interpolate the data:

x| 1|2|3|4

y| 2|1|6|27

Giuseppe Negro
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asdfnomll
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  • Did you try at all? – Stefano Mar 05 '14 at 19:44
  • @Stefano Of course. I tried use the Langrange Form for interpolating a polynomial, but I don't really understand the form too well. I was hoping by looking at an example, I will be able to understand what the method is doing. – asdfnomll Mar 05 '14 at 19:45
  • For this particular question, all you need to do is substitute $x$ and $y$ values into given polynomials and check if true. – Kaster Mar 05 '14 at 19:59
  • @Mathlovin Oh...Actually, can you tell me what "Interpolate" means? I think that is the reason I don't really understand so far – asdfnomll Mar 05 '14 at 20:13
  • @asdfnomll Interpolation means finding a polynomial that passes through given set of points. – Kaster Mar 05 '14 at 22:31

2 Answers2

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Example take the point $(x,y)=(1,2)$ and let's see if $p(x)=y$ passes through $(1,2)$:

for $x=1$ we have

$p(1)=5(1)−27(1)+45(1)−21(1)=2=y$

Then you should verify this for every point and for both function. The exercise was not asking you to interpolate the points, it was asking to verify if the provided functions do.

Stefano
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Just verify whether $Xa=y$, where $X$ is the Van Der Monde matrix: $$ X=\left(\matrix{1 & x_0 & x_0^2 & x_0^3\\ 1 & x_1 & x_1^2 & x_1^3\\ 1 & x_2 & x_2^2 & x_2^3\\ 1 & x_3 & x_3^2 & x_3^3}\right), $$ and $x_i$'s are the $x$'s that you wrote; $a$ is the polynomial coefficient vector: $$ a=(5,-27,45,-21), $$ and $y$ is the vector of the $y$'s that you wrote.

So, just compute $Xa$ and see if coincides with $y$. Repeat also for $q(x)$, i.e., choose the right $a$.

7raiden7
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