Question

If the polynomial \[ P(x)=a_0+a_1x+a_2\,x(x-1)+a_3\,x(x-1)(x-2) \] interpolates the points $(0,2)$, $(1,3)$, $(2,2)$, and $(3,5)$, then the value of $P\!\left(\tfrac{5}{2}\right)$ is ____________ (round off to 2 decimal places).

Hint:

This is the Newton interpolation form with nodes $0,1,2,3$. The coefficients $a_0,a_1,a_2,a_3$ are the successive divided differences; here they’re read off directly by plugging $x=0,1,2,3$.

Answer 1

Correct Answer: 2.6

Step 1: Use the Newton-form coefficients from data points.
At $x=0$: $P(0)=a_0=2 \Rightarrow a_0=2$.
At $x=1$: $P(1)=a_0+a_1=3 \Rightarrow a_1=1$.
At $x=2$: $P(2)=a_0+2a_1+2a_2=2 \Rightarrow 2+2(1)+2a_2=2 \Rightarrow a_2=-1$.
At $x=3$: $P(3)=a_0+3a_1+6a_2+6a_3=5 \Rightarrow 2+3-6+6a_3=5 \Rightarrow a_3=1$.
Step 2: Evaluate at $x=\dfrac{5{2}$.}
\[ \begin{aligned} P\!\left(\tfrac{5}{2}\right) &=2+1 . \tfrac{5}{2}+(-1) . \tfrac{5}{2}\!\left(\tfrac{5}{2}-1\right) +1 . \tfrac{5}{2}\!\left(\tfrac{5}{2}-1\right)\!\left(\tfrac{5}{2}-2\right)
[4pt] &=2+2.5- \Big(2.5\times 1.5\Big) + \Big(2.5\times 1.5\times 0.5\Big)
&=4.5-3.75+1.875=2.625. \end{aligned} \] Rounded to two decimals: $2.63$. Final Answer:\fbox{$2.63$}