Question

Find the missing entry in the following table

x	0	1	2	3	4
yx	1	3	9	-	81

Answer 1

Since only four values of f(x) are given the polynomial which fits the data is of degree three.

Hence fourth differences are zeros.

i.e Δ⁴y₀ = 0

(E – 1)⁴y₀ = 0

(E⁴ – 4E³ + 6E² – 4E + 1) y₀ = 0

E⁴y₀ – 4E³y₀ + 6E²y₀ – 4Ey₀ + 1y₀ = 0

y₄ – 4y₃ + 6y₂ – 4y₁+ y₀ = o

81 – 4y₃ + 6(9) – 4(3) + 1 = 0

81 – 4y₃ + 54 – 12 + 1 = 0

136 – 12 – 4y₃ = 0

4y₃ = 124

y₃ = `124/4`

∴ y₃ = 31