Question

Determine a polynomial function whose graph passes through the given points. 5: (0,-5); (1,-4) ; (-1,-9); (2,-3),

Answer 1

Answer:

The interpolating polynomial is

$p(x) = \frac{1}{2}x^3-\frac{3}{2}+2x-5$.

Step-by-step explanation:

First, notice that we want to calculate the interpolating polynomial through the points (0,-5); (1,-4) ; (-1,-9); (2,-3). This means that we want to find a polynomial $p(x)$ such that

$p(-1) = -9$, $p(0) = -5$ $p(1) = -4$ and $p(2) = -3$.

We will have four equations, so our polynomial will be, at most, of degree 3. Let us write

$p(x) = a_0 + a_1x +a_2x^2 +a_3x^3.$

The coordinates give us the following equations:

$\begin{cases} -9= p(-1) &= a_0 - a_1 +a_2 -a_3 \\ -5=p(0) &= a_0 \\-4=p(1) &= a_0 + a_1 +a_2 +a_3 \\ -3 = p(2) &= a_0 + 2a_1 +4a_2 +8a_3 \end{cases}$

Notice that from the second equation we know that $a_0 =-5$. Then, we obtained the linear system of equations

$\begin{cases} -9 &= -5 - a_1 +a_2 -a_3 \\-4 &= -5 + a_1 +a_2 +a_3 \\ -3 &= -5 + 2a_1 +4a_2 +8a_3 \end{cases}$

which is equivalent to

$\begin{cases} - a_1 +a_2 -a_3 &= -4\\ a_1 +a_2 +a_3 &= 1\\ 2a_1 +4a_2 +8a_3 &= 2\end{cases}$.

So, we have reduced our interpolation problem to solve a linear system of equations. Now, notice that if we add the first two equations of the system we obtain

$2a_2=-3$ that yields $a_2 = -\frac{3}{2}$.

Then, our system becomes

$\begin{cases}-a_1 -\frac{3}{2}-a_3 &= -4\\a_1 -\frac{3}{2} +a_3 &= 1\\ 2a_1 -6 +8a_3 &= 2\end{cases}$

which is equivalent to

$\begin{cases}-a_1-a_3 &=-\frac{5}{2}\\a_1+a_3 &=\frac{5}{2}\\ 2a_1+8a_3 &=8\end{cases}$.

Recall that now the first two equations are just the same, so we will use the first and third ones:

$\begin{cases}-a_1-a_3 &=-\frac{5}{2}\\2a_1+8a_3 &= 8\end{cases}$.

If we multiply the first one and add it to the second we get:

$6a_3 = 3$ that yields $a_3 = \frac{1}{2}$.

Thus, substituting this value in the first equation:

$-a_1-\frac{1}{2} = -\frac{5}{2}$ which is equivalent to $-a_1=-\frac{4}{2}$. Then, $a_1=2$.

Summing up all our results we get that the interpolating polynomial is

$p(x) = \frac{1}{2}x^3-\frac{3}{2}+2x-5$.